Toric geometry of path signature varieties

TL;DR

This paper explores the algebraic geometry of path signature varieties, demonstrating that they are toric in certain classes of paths, thus connecting stochastic analysis with algebraic geometry.

Contribution

It establishes that the signature varieties of rough paths and axis-parallel paths are toric, providing new geometric insights into path signatures in stochastic analysis.

Findings

Signature varieties of rough paths are toric.

Signature varieties of axis-parallel paths are toric in many cases.

Bridges algebraic geometry and stochastic analysis through toric structures.

Abstract

In stochastic analysis, a standard method to study a path is to work with its signature. This is a sequence of tensors of different order that encode information of the path in a compact form. When the path varies, such signatures parametrize an algebraic variety in the tensor space. The study of these signature varieties builds a bridge between algebraic geometry and stochastics, and allows a fruitful exchange of techniques, ideas, conjectures and solutions. In this paper we study the signature varieties of two very different classes of paths. The class of rough paths is a natural extension of the class of piecewise smooth paths. It plays a central role in stochastics, and its signature variety is toric. The class of axis-parallel paths has a peculiar combinatoric flavour, and we prove that it is toric in many cases.

Equations

$${\int }_0^1{\int }_0^{t_k}\cdots {\int }_0^{t_3}{\int }_0^{t_2}{\dot{X}}_{i_1}(t_1)\cdot \dots \cdot {\dot{X}}_{i_k}(t_k)d{t}_1\dots d{t}_k.$$

$$\sigma (X):=({\sigma }^{(k)}(X)\mid k\in \mathbb{N})$$

$$T(({\mathbb{R}}^d)):=\mathbb{R}\times {\mathbb{R}}^d\times ({\mathbb{R}}^d{)}^{\otimes 2}\times \dots$$

$$T({\mathbb{R}}^d)=\mathbb{R}⟨x_1,\dots ,x_d⟩$$

$$(X\bigsqcup Y)(t)=\{\begin{array}{ll}X(2t)& \text{mbox}ift\in \left[0,\frac{1}{2}\right],\\ X(1)+Y(2t-1)& \text{mbox}ift\in \left[\frac{1}{2},1\right].\end{array}$$

$$⟨-,-⟩:T(({\mathbb{R}}^d))\times T({\mathbb{R}}^d)\to \mathbb{R}$$

$$\displaystyle\begin{array}[]{l}e\shuffle w=w\shuffle e=w,\text{ and }\\ (w_{1}\otimes a)\shuffle(w_{2}\otimes b)=\left(w_{1}\shuffle(w_{2}\otimes b)\right)\otimes a+\left((w_{1}\otimes a)\shuffle w_{2}\right)\otimes b.\end{array}$$

$$⟨\sigma (X),v⟩\cdot ⟨\sigma (X),w⟩=⟨\sigma (X),v\text{shuffle}w⟩$$

$$\exp (T):=\sum _{n=0}^{\infty }\frac{T^{\otimes n}}{n!}.$$

$$\mathcal{G}({\mathbb{R}}^d)=\{T\in T_1(({\mathbb{R}}^d))\mid ⟨T,v⟩\cdot ⟨T,w⟩=⟨T,v\text{shuffle}w⟩\text{mbox}forallwordsv,w\in T({\mathbb{R}}^d)\}.$$

$$T^m({\mathbb{R}}^d):=\underset{k=0}{\overset{m}{⨁}}({\mathbb{R}}^d{)}^{\otimes k},$$

$$\mu(t)=\begin{cases}0&\mbox{ if $t$ is divisible by the square of a prime},\\ 1&\mbox{ if $t$ is the product of an even number of distinct primes},\\ -1&\mbox{ if $t$ is the product of an odd number of distinct primes}.\\ \end{cases}$$

$$\dim {\mathrm{Lie}}^m({\mathbb{R}}^d)=\sum _{l=1}^m\sum _{t\mid l}\frac{\mu (t)}{l}{d}^{\frac{l}{t}}.$$

$$\left|{\sigma }^{(k)}(X_{|[s,t]})\right|\text{apprle}|t-s{|}^k,$$

$$f_{d,k,m}:{\mathrm{Lie}}^m({\mathbb{R}}^d)\stackrel{\exp }{\to }\mathcal{G}({\mathbb{R}}^d)\stackrel{p_k}{\to }({\mathbb{R}}^d{)}^{\otimes k}\to ({\mathbb{R}}^d{)}^{\otimes k}\otimes \mathbb{C}=({\mathbb{C}}^d{)}^{\otimes k}⇢{\mathbb{P}}^{d^k-1}.$$

$$M_3=\left(\begin{array}{ccc}1& 10& 20\\ 16& 6& 9\\ 14& 15& 2\end{array}\right).$$

$$A=\left(\begin{tabular}[]{cccc|c}&&&&0\\ &$\lambda M_{n}$&&&0\\ &&&&$\vdots$\\ &&&&0\\ \hline\cr\bigstrut 0&$\dots$&0&0&$\lambda k_{n}$\end{tabular}\right)\mbox{ and }$$

$$B=\left(\begin{tabular}[]{cccccc|c}0&\ldots&0&0&-2n+2&2n-3&$1+x$\\ 0&\ldots&0&-2n+3&0&2n-5&$2+x$\\ $\vdots$&$\udots$&$\udots$&$\udots$&$\vdots$&$\vdots$&$\vdots$\\ 0&-n-1&0&$\dots$&0&3&$n-2+x$\\ -n&0&$\dots$&$\dots$&0&1&$n-1+x$\\ 0&0&$\dots$&$\dots$&0&$-n^{2}$&$n^{2}+x$\\ \hline\cr\bigstrut$n+x$&$n+1+x$&$\dots$&$\dots$&$2n-2+x$&$2n-1+x$&$nx-\frac{n(3n-1)}{2}$\end{tabular}\right).$$

$$\phi :\mathbb{C}[y_{\alpha }\mid \alpha \in A]\to \mathbb{C}[x_w\mid w\in W_{d,m}]$$

$$f:=y_{{\alpha }_1}\cdot \dots \cdot y_{{\alpha }_{n+1}}-y_{{\beta }_1}\cdot \dots \cdot y_{{\beta }_{n+1}}.$$

$$f=l_1(g_1-h_1)+\cdots +l_r(g_r-h_r).$$

$$f=\sum _{i=1}^k{m}_i{b}_i,$$

$$(I(2):f^{\infty }):=\{g\in R\mid g{f}^k\in I(2)\text{mbox}forsomek\in \mathbb{N}\}.$$

$$(w_{1,1},\dots ,w_{1,a_1},w_{2,1},\dots ,w_{2,a_2},\dots ,w_{m,1},\dots ,w_{m,a_m})\text{ where }\sum _{i=1}^m\sum _{j=1}^{a_i}i{w}_{i,j}=k.$$

$$f=p:=(k,0,\dots ,0)$$

$$(k-2,0,\dots ,0,1,0,\dots ,0)+(w_{1,1}+2,w_{1,2},\dots ,w_{1,a_1},w_{2,1}-1,w_{2,2}\dots ,w_{2,a_2},\dots ,w_{m,a_m}),$$

$$\left\{(t_1,\dots ,t_r)\in {\mathbb{N}}^r|\sum _{i=1}^r{w}_i{t}_i=k\right\}.$$

$$P:=P((w_1,\dots ,w_{i-1},w_i,w_i^{\prime },w_{i+1},\dots ,w_m),k),$$

$$Q:=P((w_1,\dots ,w_i,w_{i+1},\dots ,w_m),k).$$

$$\mathbb{P}(1^{a_1},\dots ,m^{a_m})⇢{\mathbb{P}}^{d^k-1}.$$

Full text

In this paper we study the signature varieties of two very different classes of paths. The class of rough paths is a natural extension of the class of piecewise smooth paths. It plays a central role in stochastics, and its signature variety is toric. The class of axis-parallel paths has a peculiar combinatoric flavour, and we prove that it is toric in many cases.

1 Introduction

A path is a continuous map $X\colon[0,1]\to\mathbb{R}^{d}$. This very simple mathematical object can be used to interpret a wide range of situations. From a physical transformation to a meteorological model, from a medical experiment to the stock market, everything that involves parameters changing with time can be described by a path. This makes paths priceless tools in many branches of mathematics, as well as in a number of applied sciences. The downside is that, being a continuous object, explicit computations on a path are not easy to handle. A typical way to overcome this problem is to find invariants that are simpler to understand and can provide us enough information.

For paths, this problem was faced in [Che54]. Assume that $X$ is piecewise smooth, and fix $k\in\mathbb{N}$. Let $X_{i}$ be the composition of $X$ with the projection to the $i$-th coordinate. Chen defined the $k$-th signature of $X$ to be the order $k$ tensor $\sigma^{(k)}(X)$ whose $(i_{1}\ldots i_{k})$-th entry is

[TABLE]

By convention, $\sigma^{(0)}(X)=1$. The sequence

[TABLE]

is called the signature of $X$. Sometimes it is also convenient to consider a truncated signature $\sigma^{\leq m}(X):=(\sigma^{(k)}(X)\mid k\in\{0,\dots,m\})$. In this context, it is natural to ask how much these tensors tell us about $X$. In [Che58, Theorem 4.1] Chen proved that, up to a mild equivalence relation, the signature allows to uniquely recover a piecewise smooth path.

Signatures are appreciated not only in stochastics, but also in topological data analysis, financial mathematics or machine learning among others (see [GLKF13, CK16, CNO18, CO18]). Chen’s iterated integrals have deep connections with de Rham homotopy theory, as shown in [Hai02]. In this paper we are interested in the geometric side, illustrated in Section 1.2. We study the so-called signature varieties – roughly speaking the geometric locus of all possible signatures. In Section 2 we consider the variety $\mathcal{R}_{d,k,m}$ of signatures of rough paths. Our results often rely on toric geometry, which allows us to answer several questions about their equations and singularities:

• •

Answering a question raised in [AFS18] and [Gal18], we show that the ideal of $\mathcal{R}_{d,k,m}$ may be generated in arbitrary high degree - Proposition 2.9. However, we prove that quadratic polynomials define a (possibly reducible) variety, of which $\mathcal{R}_{d,k,m}$ is an irreducible component - Proposition 2.11.

• •

We characterize the cases in which $\mathcal{R}_{d,k,m}$ is an embedding of the weighted projective space. We find conditions that make it (projectively) normal and examples when it is not - Subsection 2.3.

• •

We study the asymptotic behavior of the degree of $\mathcal{R}_{d,k,m}$ and we give explicit formulas for some special cases - Subsection 2.4.

In Section 3 we study the geometry of the signature variety $\mathcal{A}_{\nu,k}$ of axis parallel paths:

• •

We provide an easy, combinatorial parametrization of the variety - Lemma 3.3.

• •

We apply the above description to prove that $\mathcal{A}_{\nu,k}$ is toric in several cases - Section 3.1. We study the dimension of $\mathcal{A}_{\nu,k}$ exhibiting defective cases - Section 3.2.

Finally, we use our knowledge of axis paths to prove a general formula for the determinant of the signature matrix of any path, Theorem 3.19 and Corollary 3.23.

In order to present those results we need to recall the algebraic background.

1.1 The tensor algebra

The $k$-th signature of a path $X$ belongs to $(\mathbb{R}^{d})^{\otimes k}$. We now introduce an ambient space for the whole signature $\sigma(X)$.

Definition 1.1**.**

The tensor algebra over $\mathbb{R}^{d}$ is the graded $\mathbb{R}$-vector space

[TABLE]

of formal power series in the non-commuting variables $x_{1},\dots,x_{d}$. It is an $\mathbb{R}$-algebra with respect to the tensor product. The algebraic dual of $T((\mathbb{R}^{d}))$ is the graded free $\mathbb{R}$-algebra

[TABLE]

of polynomials in the non-commuting variables $x_{1},\dots,x_{d}$.

These spaces and their rich algebraic structures are well studied. In this section we recall the features we need, and we refer to [Reu93] for a detailed treatment. Next result, proven in [Che57, Section 2], gives a first taste of the correlation between the tensor algebra and signatures of paths. Recall that the concatenation of two paths $X$ and $Y$ is the path $X\sqcup Y\colon[0,1]\longrightarrow\mathbb{R}^{d}$ given by

[TABLE]

Lemma 1.2** (Chen’s identity).**

If $X,Y\colon[0,1]\to\mathbb{R}^{d}$ are piecewise smooth paths, then $\sigma(X\sqcup Y)=\sigma(X)\otimes\sigma(Y)$ as formal power series in $T((\mathbb{R}^{d}))$.

We now introduce a useful notation.

Notation 1.3**.**

For $T\in T((\mathbb{R}^{d}))$, denote by $T_{i_{1}\dots i_{k}}$ the $(i_{1}\dots i_{k})$-th entry of the order $k$ element of $T$. For $y\in\mathbb{R}$, we set $T_{y}((\mathbb{R}^{d})):=\{T\in T((\mathbb{R}^{d}))\mid T_{0}=y\}$ to be the space of tensor sequences with constant entry 0. Moreover, we will identify a degree $k$ monomial $x_{i_{1}}\cdot\ldots\cdot x_{i_{k}}$ with the word $w=i_{1}\ldots i_{k}$ in the alphabet $\{1,\dots,d\}$. The number $k$ is called the length of $w$ and it is denoted by $\ell(w)$. The degree 0 monomial corresponds to the empty word $e$. In this way, the tensor product of the two monomials corresponding to the words $v$ and $w$ is the word obtained by writing $v$ followed by $w$, and it is called the concatenation product. The natural duality pairing

[TABLE]

is given by $\langle T,i_{1}\ldots i_{k}\rangle=T_{i_{1}\dots i_{k}}$, and extended by linearity.

Besides the concatenation of words, there is another product on $T(\mathbb{R}^{d})$. It has a strong combinatoric flavour, but it will also allow us to define very interesting algebraic objects.

Definition 1.4**.**

We define the shuffle product of two words recursively by

[TABLE]

Less formally, $v\shuffle w$ is the sum of all order-preserving interleavings of $v$ and $w$.

For instance, $1\shuffle 23=123+213+231$. Despite its apparently complicated definition, the shuffle product enjoys good properties. For instance, $(T(\mathbb{R}^{d}),\shuffle,e)$ is a commutative algebra. Moreover next Lemma, proven in [Reu93, Proof of Corollary 3.5], shows that the shuffle product behaves nicely with respect to the signatures.

Lemma 1.5** (Shuffle identity).**

If $X$ is a piecewise smooth path, then

[TABLE]

for all words $v,w\in T(\mathbb{R}^{d})$.

It follows that signatures do not fill the tensor space $T((\mathbb{R}^{d}))$, but rather they live in a subset. In order to make this observation precise, we introduce an important subspace of the tensor algebra.

Definition 1.6**.**

Consider the Lie bracketing $[T,S]=TS-ST$ on $T((\mathbb{R}^{d}))$. We define $\mathop{\rm Lie}\nolimits(\mathbb{R}^{d})\subset T_{0}((\mathbb{R}^{d}))$ to be the free Lie algebra generated by $x_{1},\dots,x_{d}$, that is, the smallest vector subspace of $T((\mathbb{R}^{d}))$ that contains $x_{1},\dots,x_{d}$ and is closed with respect to the bracketing.

The Lie group associated to $\mathop{\rm Lie}\nolimits(\mathbb{R}^{d})$ is an important object. One way to define it is as the image of $\mathop{\rm Lie}\nolimits(\mathbb{R}^{d})$ under the exponential map.

Definition 1.7**.**

Define $\exp:T_{0}((\mathbb{R}^{d}))\to T_{1}((\mathbb{R}^{d}))$ by the formal power series

[TABLE]

We denote $\mathcal{G}(\mathbb{R}^{d}):=\exp(\mathop{\rm Lie}\nolimits(\mathbb{R}^{d}))$.

By construction, $(\mathcal{G}(\mathbb{R}^{d}),\otimes,e)$ is a group. However, there is another way to characterize it in terms of shuffle product. By [Reu93, Theorem 3.2],

[TABLE]

Now we see the first clear connection with the signatures. By the shuffle identity, $\mathcal{G}(\mathbb{R}^{d})$ contains the signatures of all piecewise smooth paths.

For our purposes, we need to point out that every definition we recalled has a truncated version. Namely, one can fix $m\in\mathbb{N}$ and consider

[TABLE]

where tensors of order greater than $m$ are set to zero. Inside $T^{m}(\mathbb{R}^{d})$ there are $\mathcal{G}^{m}(\mathbb{R}^{d})$ and $\mathop{\rm Lie}\nolimits^{m}(\mathbb{R}^{d})$. The restriction of the map $\exp$ is defined in the same way, and we will write $\exp^{(m)}$ to denote the map $T_{0}^{m}((\mathbb{R}^{d}))\to T_{1}^{m}((\mathbb{R}^{d}))$.

Another feature of the group $\mathcal{G}(\mathbb{R}^{d})$ is that, for every $m\in\mathbb{N}$, $\mathcal{G}^{m}(\mathbb{R}^{d})$ not only contains, but actually coincides with the set of all truncated signatures of smooth paths (see [AFS18, Theorem 4.4]) and it is a Lie group (see [FV10, Theorem 7.30]). Moreover, it is defined by finitely many polynomials - namely, the shuffle relations with words of length at most $m$ - hence it is an algebraic variety. It is irreducible by [AFS18, Theorem 4.10]. On the other hand, $\mathop{\rm Lie}\nolimits^{m}(\mathbb{R}^{d})$ is a finite dimensional vector space. In order to provide it with a basis, it is time to introduce another powerful combinatoric concept.

Definition 1.8**.**

A non-empty word $w$ is Lyndon if, whenever we write $w=pq$ as the concatenation of two nonempty words, we have $w<q$ in the lexicographic order. In this case there is a unique pair $(p,q)$ of nonempty words such that $w=pq$ and $q$ is minimal with respect to lexicographic order. The bracketing of $w$ is $[p,q]=pq-qp$.

As shown in [Reu93, Corolllary 4.14] and [AFS18, Proposition 4.7], the bracketings of all Lyndon words of length at most $m$ form a basis for $\mathop{\rm Lie}\nolimits^{m}(\mathbb{R}^{d})$. Therefore, in order to compute its dimension - and thus the dimension of the variety $\mathcal{G}^{m}(\mathbb{R}^{d})$ - it is enough to count Lyndon words. First, recall that the Möbius function $\mu:\mathbb{N}\to\mathbb{N}$ sends a natural number $t$ to

[TABLE]

Then the number of Lyndon words of length $l$ in the alphabet $\{1,\dots,d\}$ is $\sum_{t\mid l}\frac{\mu(t)}{l}d^{\frac{l}{t}},$ and therefore

[TABLE]

1.2 Signatures from a geometric viewpoint

Chen’s theorem assumes the knowledge of the $k$-th signature for every $k\in\mathbb{N}$. What happens when we know only one of them? In [AFS18], the authors consider the problem from an algebraic geometry perspective. If we fix a certain class of paths and the order $k$ of the tensors, then the $k$-th signature $\sigma^{(k)}$ is an algebraic map into $(\mathbb{R}^{d})^{\otimes k}$. The closure of the image of $\sigma^{(k)}$ is called the $k$-signature variety. We get a first example by considering the class of all smooth paths in $\mathbb{R}^{d}$. If $X$ is smooth, then the $k$-th signature of $X$ is just the $k$-th entry of $\sigma(X)\in\mathcal{G}(\mathbb{R}^{d})$. This leads to the following definition.

Definition 1.9**.**

The universal variety $\mathcal{U}_{d,k}\subset(\mathbb{R}^{d})^{\otimes k}$ is the projection of $\mathcal{G}(\mathbb{R}^{d})$ onto the $k$-th factor. It is the closure of the set of all $k$-th signatures of smooth paths.

Since every $\mathcal{G}^{m}(\mathbb{R}^{d})$ is an irreducible variety, $\mathcal{U}_{d,k}$ is irreducible as well. By [AFS18, Theorem 6.1], its dimension coincides with $\dim\mathcal{G}^{k}(\mathbb{R}^{d})$. Equivalently, we can compute it as the dimension of the vector space $\mathop{\rm Lie}\nolimits^{k}(\mathbb{R}^{d})$, that is the number of Lyndon words of length at most $k$ in the alphabet $\{1,\dots,d\}$.

This geometric approach provides a way to translate questions about paths into questions about the signature variety. For instance, the crucial problem of reconstructing uniquely the path from its $k$-signature translates into the injectivity of $\sigma^{(k)}$. This leads to study the fibers of $\sigma^{(k)}$ and therefore the dimension of the signature variety. Provided that a preimage can be reconstructed, such computation consists in solving a system of polynomial equations, and we can get an idea of how difficult that is by looking at the degree of the variety or at the degrees of the generators of its ideal. Taking a step further, the singular locus can point out that some signatures are different from the others, so we have a meaningful way to distinguish special paths in our class.

There are many interesting questions when it comes to study a signature variety, and we do not hope to give full answers to all of them. Path recovery from the third signature is the main goal of [PSS18]. Further interplay between the tensor algebra and the signatures is explored in [CP18]. Our contribution to this topic is a detailed study of the rough paths signature variety, presented in Section 2, as well as the analysis of the axis parallel signature variety, which we describe in Section 3. Both varieties present several geometric subtleties. When dealing with them, we could take advantage of a surprisingly rich combinatorial structure, that allows us to integrate a projective geometric approach with more computational techniques.

2 The rough Veronese variety

Researchers in stochastic analysis usually work with paths that are not piecewise smooth, and therefore do not have a signature in the sense of Chen’s definition. One of the most important examples is the class of rough paths. The interest in rough paths is quickly growing. Applications include the study of controlled ODEs and stochastic PDEs (see [LCL07]) as well as sound compression (see [LS05]), not to mention mathematical physics (see [FGL15]). The main references on rough paths are [FV10] and [FH14]. In this section we recall what the signature of a rough path is, and study the corresponding signature variety.

The signature variety of all rough paths is the whole universal variety. In the same way as polynomial paths are an interesting subclass of smooth paths, in [AFS18, Section 5.4] the authors consider a nice subclass of rough paths, parametrized by $\mathop{\rm Lie}\nolimits^{m}(\mathbb{R}^{d})$. Their signature variety $\mathcal{R}_{d,k,m}$ exhibits similarities with the classical Veronese variety and it was therefore named the rough Veronese variety - see Definition 2.3. Another reason to study $\mathcal{R}_{d,k,m}$ is that we can deduce many properties of the universal varieties from it. Indeed, $\mathcal{U}_{d,k}=\mathcal{R}_{d,k,m}$ for every $m\geq k$ and moreover, $\mathcal{U}_{d,k}=\mathcal{R}_{d,k,k}$ is a cone over $\mathcal{R}_{d,k,k-1}$ (see [Gal18, Proposition 26]).

2.1 Preliminaries

One purpose of the definition of a rough path is to generalize the concept of smooth path. We consider then a smooth path $X$, and without loss of generality we assume $X(0)=0$. Fix $t\in[0,1]$. In the definition of $k$-th signature we can replace the integral on $[0,1]$ with an integral on $[0,t]$. This is the same as restricting $X$ to $[0,t]$, hence we will denote such integral by $\sigma^{(k)}(X_{|[0,t]})$. For every $k$, we notice that $\sigma^{(k)}(X_{|[0,t]})$, as a function of $t$, is a path $[0,1]\to(\mathbb{R}^{d})^{\otimes k}$. If we look at the full signature $\sigma(X_{|[0,t]})$, we get a path $[0,1]\to\mathcal{G}(\mathbb{R}^{d})$. Notice that the signature of $X$ is the endpoint of such path. By [Gal18, Lemma 10], this $\mathcal{G}(\mathbb{R}^{d})$-valued path satisfies the Hölder-like inequality

[TABLE]

where $f(x)\apprle g(x)$ means that there is a constant $c$ such that $f(x)\leq c\cdot g(x)$ for every $x$. Summing up, a smooth path $X\colon[0,1]\to\mathbb{R}^{d}$ induces a path $\sigma(X_{|[0,\cdot]})\colon[0,1]\to\mathcal{G}(\mathbb{R}^{d})$ satisfying inequality (2). If we want a rough path to be a generalization of a smooth path, we can use this property, also allowing different exponents. Let $p_{k}:T((\mathbb{R}^{d}))\to(\mathbb{R}^{d})^{\otimes k}$ be the projection.

Definition 2.1**.**

A rough path of order $m$ is a path ${\bf X}\colon[0,1]\to\mathcal{G}^{m}(\mathbb{R}^{d})$ such that $|p_{k}({\bf X}(s)^{-1}\otimes{\bf X}(t))|\apprle|t-s|^{\frac{k}{m}}$ for every $k\in\{1,\dots,m\}$ and every $s,t\in[0,1]$. The inverse is taken in the group $\mathcal{G}^{m}(\mathbb{R}^{d})$.

As we anticipated, we will focus on a special subclass of rough paths of order $m$, indexed by elements $L\in\mathop{\rm Lie}\nolimits^{m}(\mathbb{R}^{d})$.

Definition 2.2**.**

For $L\in\mathop{\rm Lie}\nolimits^{m}(\mathbb{R}^{d})$, consider the path ${\bf X}_{L}\colon[0,1]\to\mathcal{G}^{m}(\mathbb{R}^{d})$ sending $t$ to $\exp^{(m)}(tL)$. By [FV10, Exercise 9.17], this is indeed an order $m$ rough path, and we define its signature to be its endpoint $\sigma({\bf X}_{L}):=\exp(L)\in\mathcal{G}(\mathbb{R}^{d})$.

We want to parametrize the variety containing all $\sigma^{(k)}({\bf X}_{L})$ when $L$ ranges in $\mathop{\rm Lie}\nolimits^{m}(\mathbb{R}^{d})$, so we are interested in the image of $p_{k}\circ\exp:\mathop{\rm Lie}\nolimits^{m}(\mathbb{R}^{d})\to(\mathbb{R}^{d})^{\otimes k}$. While $\exp$ is not an algebraic map, $p_{k}\circ\exp$ is. In general, the image of an algebraic map is only a semialgebraic subset of the real affine space $(\mathbb{R}^{d})^{\otimes k}$. However, it is simpler to work with complex projective varieties, so we will follow a common approach in applied algebraic geometry and consider the Zariski closure of the image, take the complexification and pass to the projectivization.

Definition 2.3**.**

The rough Veronese variety $\mathcal{R}_{d,k,m}$ is the closure of the image of the composition

[TABLE]

In general it is not easy to work out all the invariants of a given variety. Luckily, we will shortly see that the rough Veronese variety is toric. In other words it is the closure of the image of a monomial map. There are a lot of tools and techniques that make toric varieties accessible and easy to work with. Classical references on toric varieties are [CLS11, Stu96]. First, we fix some notation.

Notation 2.4**.**

For every $d\geq 2$, let $W_{d,m}$ be the set of Lyndon words of length at most $m$ in the alphabet with $d$ letters. Let $S_{d,m}:=\mathbb{C}[x_{w}\mid w\in W_{d,m}]$ be a polynomial ring with as many variables as Lyndon words. On this algebra we define a grading by setting the weight of $x_{w}$ to be the length of $w$.

Let $A$ be the set of all monomials in $\mathbb{C}[x_{w}\mid w\in W_{d,m}]$ of weighted degree $k$. Define another polynomial ring $R_{d,k}:=\mathbb{C}[y_{\alpha}\mid\alpha\in A]$ with as many variables as elements of $A$, and the usual polynomial grading.

Our starting point will be the following result, proven in [Gal18, Proposition 19]. Not only we know that $\mathcal{R}_{d,k,m}$ is toric, but we also know what are the monomials parametrizing it.

Theorem 2.5**.**

Up to projectivization, $\mathcal{R}_{d,k,m}$ is isomorphic to the closure of the image of the map $\operatorname{Spec}S_{d,m}\rightarrow\operatorname{Spec}R_{d,k}$ given by all monomials of weighted degree $k$. That is, the kernel of the map $R_{d,k}\rightarrow S_{d,m}$ is the homogeneous prime ideal defining $\mathcal{R}_{d,k,m}$.

More precisely, it is the image of a weighted projective space by the map given by all sections of $\mathcal{O}(k)$, i.e. all monomials of (weighted) degree $k$. Such a map does not have to be an embedding. In fact, $\mathcal{O}(k)$ does not need to be a Cartier divisor.

Remark 2.6**.**

The classical $k$-Veronese variety is the image of the map given by all degree $k$ monomials in the usual grading. Theorem 2.5 shows that $\mathcal{R}_{d,k,m}$ can be seen as a weighted version of the Veronese variety, thereby justifying its name.

Theorem 2.5 allows us to investigate $\mathcal{R}_{d,k,m}$ with the tools of toric geometry. In some of our combinatoric arguments, we will find it convenient to use the following notation.

Notation 2.7**.**

For any integers $a,b$ we denote by $(a^{b})$ a sequence or multiset of $b$ elements equal to $a$. For example $(1^{3},2^{2})=(1,1,1,2,2)$.

We start by describing the ideal of this variety, i.e. the polynomials that vanish on it.

2.2 Polynomials defining the rough Veronese variety

The classical Veronese variety is defined by quadrics, so it is natural to ask whether $\mathcal{R}_{d,k,m}$ enjoys the same property. Despite holding in many examples (see [AFS18, Section 4.3] and the computations in [Gal18]), the conjecture that $\mathcal{R}_{d,k,m}$ is always defined by quadrics is false. In [Gal18, Proposition 28] we see that the ideal $\mathcal{R}_{2,20,14}$ has a cubic generator. One could still hope to bound the degree of the generators. Surprisingly no such bound exists, as we will shortly prove. First we need the following lemma from linear algebra.

Lemma 2.8**.**

For every $n\geq 3$, there exist $k_{n}\in\mathbb{N}$ and an $n\times n$ matrix $M_{n}$ with positive integer entries satisfying the following properties:

the sum of the entries of each row and each column of $M_{n}$ equals $k_{n}$, 2. 2.

any subset of $n$ entries of $M_{n}$ that sum up to $k_{n}$ is either a row or a column; 3. 3.

All entries of $M_{n}$ are distinct; in particular, each column, is distinct from each row.

We refer to the matrix $M_{n}$ as a rigid $k_{n}$-square matrix.

Proof.

We proceed by induction on $n$. If $n=3$, we take $k_{3}=31$ and

[TABLE]

Assume the statement holds for $n$ and let us prove it for $n+1$. By induction hypothesis, there exist $k_{n}\in\mathbb{N}$ and an $n\times n$ matrix $M_{n}$ with the required properties. Let $\lambda\gg x\gg n$ be natural numbers and define $k_{n+1}=\lambda k_{n}+x$. Moreover, define the $(n+1)\times(n+1)$ matrices

[TABLE]

[TABLE]

Then we take $M_{n+1}:=A+B$. The only nontrivial property we have to show is point $2$. If a given subset $S$ of entries contains the lower right entry, then it may only contain the elements from the last row or column, as otherwise the sum of elements would exceed $k_{n+1}$. If it additionally contains $n^{2}+x$, then it must contain only the smallest elements left: that is those from the last column. If it does not contain $n^{2}+x$ then it must contain the largest elements left: those from the last row.

Hence, from now on we assume that $S$ does not contain the $(n+1,n+1)$ entry of $M_{n+1}$. As $\sum_{s\in S}s=x$ modulo $\lambda$ and $x\gg n$ we see that $S$ must contain precisely one element from last row or column. As the sum of $n$ elements of the upper left $n\times n$ submatrix of $B$ is much larger then $-\lambda$ and much smaller than $\lambda$ we see that the corresponding elements of $A$ must sum to $\lambda k_{n}$. By inductive assumption $S$ must contain a row or a columns of $A$. Each such row or column can only be uniquely completed to a row or column of $M_{n+1}$, as all the entries in the last row and column are distinct. ∎

Thanks to Lemma 2.8, we are now able to produce instances of $\mathcal{R}_{d,k,m}$ generated in arbitrarily high degree.

Proposition 2.9**.**

For every $n\geq 3$ and every $d\geq 2$, there exist $k,m\in\mathbb{N}$ such that $k\geq m$ and the ideal of $\mathcal{R}_{d,k,m}$ is not generated in degree $n$.

Proof.

By Lemma 2.8, there exists $k\in\mathbb{N}$ and a rigid $k$-square matrix $M$ of size $(n+1)\times(n+1)$. Let $m$ be the largest entry of $M$. By Theorem 2.5, the ideal $I$ of $\mathcal{R}_{d,k,m}$ is the kernel of the ring homomorphism

[TABLE]

sending $y_{\alpha}$ to the weighted monomial $\alpha$.

For every $i\in\{1,\dots,m\}$, fix once and for all a variable of weight $i$. This means that we fix a length $i$ Lyndon word $w$ and we associate to $i$ the variable $x_{w}$. For every $j\in\{1,\dots,n+1\}$, consider the monomial $\alpha_{j}$ defined by the product of $n+1$ variables $x_{w}$ whose weights are the entries of the $j$-th row of $M$. By construction, this monomial has weighted degree $k$ and therefore $\alpha_{1},\dots,\alpha_{n+1}\in A$. In a similar way, we can consider the monomials $\beta_{1},\dots,\beta_{n+1}$ defined by the columns of $M$. We define

[TABLE]

This is a degree $n+1$ element of $\mathbb{C}[y_{\alpha}\mid\alpha\in A]$. Moreover, since the degrees appearing in the rows of $M$ are the same as the degrees appearing in the columns, $f\in\ker(\varphi)=I$. Let us show that $f$ is not generated by degree $n$ elements. Let $q_{1},\dots,q_{r}$ be the polynomials spanning the degree $n$ part of $I$. Since $\mathcal{R}_{d,k,m}$ is toric, we can assume these are binomials [CLS11, Proposition 1.1.9]. Hence, $q_{i}=g_{i}-h_{i}$, where each $g_{i}$ and each $h_{i}$ is a degree $n$ monomial. Suppose by contradiction that $f$ can be generated by $q_{1},\dots,q_{r}$. Then there exist variables $l_{1},\dots,l_{r}$ such that $f$ is the sum

[TABLE]

Without loss of generality, assume that $y_{\alpha_{1}}\cdot\ldots\cdot y_{\alpha_{n+1}}=l_{1}g_{1}$. Since $l_{1}g_{1}-l_{1}h_{1}\in I$, we know that $\varphi(l_{1}h_{1})=\varphi(l_{1}g_{1})$. As a multiset, the variables in $\varphi(l_{1}h_{1})$ are the entries of the matrix $M$. Furthermore, the variables of $l_{1}h_{1}$ provide a partition into $n+1$ sets of cardinality $n+1$, each with sum $k$. Since $h_{1}\neq g_{1}$, by assumption on $M$, this partition must correspond to the column partition of $M$. Hence, $l_{1}(h_{1}-g_{1})=f$. This is a contradiction, because no variable divides $f$ by construction. ∎

In applications, sometimes one has a signature coming from experiments and wants to know whether it belongs to a path of a certain class. In other words, we have a point in $\mathcal{U}_{d,k}$ and we want to understand if it belongs to a given signature subvariety. From this viewpoint Proposition 2.9 may seem disappointing. In order to check whether a given point belongs to $\mathcal{R}_{d,k,m}$, we might have to evaluate polynomials of very high degree. The good news is that in most cases it is enough to only evaluate quadrics. More precisely, $\mathcal{R}_{d,k,m}$ is always defined by quadrics outside of a coordinate linear subspace of large codimension. In order to prove that, we recall a classical lemma from toric geometry.

Lemma 2.10**.**

Let $s\in\mathbb{N}$ and let $I$ be the homogeneous toric ideal associated to a set of lattice points $A\subset\mathbb{Z}^{n}$. Let $g$ be a binomial which we write as $p_{1}+\ldots+p_{t}-(p_{1}^{\prime}+\ldots+p_{t}^{\prime})$, where $p_{i},p_{j}^{\prime}\in A$. Then $g\in I(s)$ if and only if starting from the multiset $\{p_{i}\}$ we can reach the multiset $\{p_{j}^{\prime}\}$ in finitely many steps of the following form. In each step we replace $p_{i_{1}},\ldots,p_{i_{s}}$ with some other $r_{i_{1}},\ldots,r_{i_{s}}\in A$ such that $p_{i_{1}}+\ldots+p_{i_{s}}=r_{i_{1}}+\ldots+r_{i_{s}}$.

Proof.

A toric ideal is known to be binomial. Thus, $f\in I(s)$ if and only if:

[TABLE]

where $m_{i}$ are monomials and $b_{i}\in I$ are binomials of degree at most $s$. The proof is by induction on $k$. The monomial of $f$ corresponding to $p_{1}+\dots+p_{t}$ must appear on the right hand side, say in $m_{1}b_{1}$. Then $f-m_{1}b_{1}$ is obtained by applying one step of the procedure described in the lemma. We conclude by induction. ∎

Proposition 2.11**.**

Fix $R=R_{d,k}$ as in Notation 2.4. Let $I\subset R$ be the ideal of $\mathcal{R}_{d,k,m}$ and let $I(2)\subset I$ be the vector subspace of degree two forms. Let $V\subset\mathbb{P}^{d^{k}-1}$ be the variety defined by $I(2)$. Then $\mathcal{R}_{d,k,m}$ is an irreducible component of $V$.

As we will stress in Remark 2.12, if there are other components, then they are contained in a small subspace of $\mathbb{P}^{d^{k}-1}$.

Proof.

Since $I(2)\subset I$, it is clear that $\mathcal{R}_{d,k,m}\subset V$. We want to find a polynomial $f\in R\setminus I$ such that the saturation of $I(2)$ with respect to $f$ is $I$. Recall that such saturation is defined by

[TABLE]

So we want to prove that there exists an $f$ such that $(I(2):f^{\infty})=I$. This implies that $I$ and $I(2)$ coincide in the localization $R_{f}$ and therefore $V$ and $\mathcal{R}_{d,k,m}$ coincide in the affine open subset $(f\neq 0)$. In particular, $\mathcal{R}_{d,k,m}$ is an irreducible component of $V$.

Let $\mu=\dim\mathop{\rm Lie}\nolimits^{m}(\mathbb{R}^{d})$ and let $A\subset\mathbb{N}^{\mu}$ be the set of lattice points associated to the monomials defining the toric variety $\mathcal{R}_{d,k,m}$. By Theorem 2.5, points of $A$ are of the form

[TABLE]

By [CLS11, Proposition 1.1.9], there exists a set of binomial generators of $I$. Each of such binomials corresponds to an integral relation $p_{1}+\ldots+p_{t}=p_{1}^{\prime}+\ldots+p_{t}^{\prime}$, where all $p_{i}$ and $p_{i}^{\prime}$ are in $A$ (the sum of points corresponds to the product of variables). Our task is to find $f\notin I$ such that $f^{n}g\in I(2)$ for every generator $g=p_{1}+\ldots+p_{t}-p_{1}^{\prime}-\ldots-p_{t}^{\prime}$ of $I$ for some $n$. We will define $f$ to be a variable. In toric words, $f$ corresponds to a lattice point $p\in A$. By Lemma 2.10, we want to find $p\in A$ such that $p+\ldots+p+p_{1}+\ldots+p_{t}$ can be turned in $p+\ldots+p+p_{1}^{\prime}+\ldots+p_{t}^{\prime}$ by repeatedly replacing a pair of summands with another pair having the same sum. Let

[TABLE]

and $p_{1}=(w_{1,1},\ldots,w_{1,a_{1}},w_{2,1},\ldots,w_{2,a_{2}},\ldots,w_{m,1},\ldots,w_{m,a_{m}})$. Assuming $w_{2,1}>0$, we can replace $p+p_{1}$ by

[TABLE]

i.e. we replace two on the first coordinate with one on the coordinate corresponding to $w_{2,1}$. By iterating this process, we add more copies of $p$ so that we can replace $p+\ldots+p+p_{1}$ by points for which at most one coordinate $w_{i,j}$ is nonzero for $(i,j)\neq(1,1)$, and that coordinate is equal to one. We do the same to $p_{2},\dots,p_{t}$, so that we can replace $p+\ldots+p+p_{1}+\ldots+p_{t}$ by a sum of points for which at most one coordinate $w_{i,j}$ is nonzero for $(i,j)\neq(1,1)$, and that coordinate is equal to one. We apply the same process to $p_{1}^{\prime}+\ldots+p_{t}^{\prime}$. Since now both sides are broken into this kind of simple pieces, and since the condition of having weighted sum $k$ is always preserved, the only possibility is that the summands are pairwise the same. ∎

Remark 2.12**.**

In the proof of Proposition 2.11 we could have chosen the point $p$ to be any point with coordinates $w_{i,j}=0$ for $i>1$. This proves that $I(2)$ and $I$ define the same scheme (in particular, the quadrics define the correct set) outside of the locus where all these coordinates vanish. Therefore the other components of the variety defined by $I(2)$ (embedded or not) must be supported on a coordinate subspace of large codimension.

2.3 Normality of the rough Veronese Variety

A classical approach to study geometry of a projective toric variety is to look at the associated lattice polytope [CLS11, Stu96, Ful93]. In our case the central object is presented in the following definition.

Definition 2.13**.**

For $k\in\mathbb{N}$ and $w=(w_{1},\dots,w_{r})\in\mathbb{N}^{r}$, we define the lattice polytope $P(w,k)$ as the convex hull of

[TABLE]

In this notation, the polytope associated to $\mathcal{R}_{d,k,m}$ is $P((1^{a_{1}},2^{a_{2}},\dots,m^{a_{m}}),k)$, where $a_{i}$ is the number of Lyndon words of length $i$ in the alphabet $\{1,\dots,d\}$. Many authors, like [CLS11, Ful93], require a toric variety to be normal. This is equivalent to the fact that the toric variety may be represented by a fan. Hence, the first important task is to decide when $\mathcal{R}_{d,k,m}$ is normal. One advantage is that normality can be checked on the polytope.

Lemma 2.14**.**

A polytope $P$ is normal if and only if $kP$ is normal for every $k\in\mathbb{N}$. Moreover, if $X_{P}$ is the associated toric variety, then

$X_{P}$* is normal if and only if $P$ is very ample,* 2. 2.

$X_{P}$* is projectively normal (i.e. the affine cone over it is normal) if and only if $P$ is normal.*

Proof.

These results are proven in [CLS11, Chapter 2], together with many other features of normal polytopes. ∎

Lemma 2.14 will help us to determine when $\mathcal{R}_{d,k,m}$ is normal. We start with the following result, that simplifies the problem.

Lemma 2.15**.**

$P((w_{1}^{a_{1}},w_{2}^{a_{2}},\dots,w_{m}^{a_{m}}),k)$* is normal if and only if $P((w_{1},w_{2},\dots,w_{m}),k)$ is normal. In particular, the normality of $\mathcal{R}_{d,k,m}$ does not depend on $d$.*

Proof.

By induction, it is enough to check what happens when we add or discard one of the entries. In order to simplify notation, set

[TABLE]

where $w_{i}=w_{i}^{\prime}$ and we do not assume that the $w_{j}$’s are distinct. Observe that one of the facets of $P$ is precisely the polytope

[TABLE]

Since every face of a normal polytope is normal, the first implication follows.

Assume now that $Q$ is normal. Consider the linear surjection $\sim:P\to Q$, defined by the sum of the two entries corresponding to $w_{i}$ and $w_{i}^{\prime}$. If $t=(t_{1},\ldots,t_{i-1},t_{i},t_{i}^{\prime},t_{i+1},\dots,t_{m})\in P$, then $\tilde{t}=(t_{1},\ldots,t_{i-1},t_{i}+t_{i}^{\prime},t_{i+1},\dots,t_{m})\in Q$. Note that $\sim$ is also surjective on lattice points, that is $\sim:P\cap\mathbb{N}^{r}\to Q\cap\mathbb{N}^{r-1}$ is surjective. Let $p$ be a lattice point in some multiple $sP$ of $P$. We can write $p=\lambda_{1}p_{1}+\ldots+\lambda_{s}p_{s}$, where $p_{1},\dots,p_{s}\in P$ are lattice points and $\lambda_{1}+\ldots+\lambda_{s}=s$. Since $\tilde{p}\in sQ$, $Q$ is normal by hypothesis and $\sim$ is surjective on lattice points, there are $x_{1},\ldots,x_{s}\in P\cap\mathbb{N}^{r}$ such that $\tilde{p}=\lambda_{1}\tilde{p}_{1}+\ldots+\lambda_{s}\tilde{p}_{s}=\tilde{x}_{1}+\ldots+\tilde{x}_{s}$. Let $x_{i}\in P$ be the preimage of $\tilde{x}_{i}$ having 0 in the entry corresponding to $w_{i}^{\prime}$, and let $x=x_{1}+\ldots+x_{s}$. Since $\tilde{p}=\tilde{x}$, $p$ and $x$ coincide on every coordinate except the ones corresponding to $w_{i}$ and $w_{i}^{\prime}$. Moreover, these entries have the same sum, and such sum is an integer by construction of $x$. Then it is enough to increase the zero entry of some suitable $x_{i}$, keeping such sum untouched, to obtain from $x_{1},\dots,x_{s}$ another set of $s$ lattice points of $P$ whose sum is $p$. Therefore $P$ is normal.∎

In general, the rough Veronese variety does not need to be normal. As we show using the code available online [CGM], $P((1,2,\ldots,9),18)$ is not very ample, so $\mathcal{R}_{d,18,9}$ is not normal for any $d\geq 2$.

In algebraic geometry, given a map $f=(f_{1},\dots,f_{s})$ defined by homogeneous polynomials of the same degree, it is natural to study the induced rational map $\tilde{f}$ between projective spaces. The map $\tilde{f}$ may be not defined everywhere - the locus where all of the polynomials $f_{i}$ vanish is called the base locus or indeterminacy locus of $\tilde{f}$. In our setting, the monomials defining $\mathcal{R}_{d,k,m}$ are of degree $k$, however in variables that are of different degrees. This implies that instead of considering the classical projective space, we need the weighted projective space. Theorem 2.5 tells us that $\mathcal{R}_{d,k,m}$ is the closure of the image of a rational map

[TABLE]

The codomain is the usual projective space and the domain is the weighted projective space with $\sum_{i}a_{i}$ variables of weights as given in the brackets. The map is given by all monomials of degree $k$.

Before we proceed further, let us recall that maps from an algebraic variety $X$ to a projective space are studied through line bundles (or equivalently Cartier divisors) on $X$ [Laz04]. The theory of line bundles on weighted projective spaces is very well understood. The Picard group equals $\mathbb{Z}$ where $\bf{1}$ is an ample generator. The global sections of this generator may be identified with all monomials of degree $l$ equal to the least common multiple of all the degrees appearing in the weighted projective space. Thus, it is customary to denote it by $\mathcal{O}(l)$. In particular, the elements of the Picard group will be denoted by $\mathcal{O}(s)$, for $s$ divisible by $l$. Although the class group is abstractly also equal to $\mathbb{Z}$, it is larger. The inclusion of the Picard group in the class group $\mathbb{Z}\hookrightarrow\mathbb{Z}$ is given by multiplication by $l$. In particular, the elements of the class group will be denoted by $\mathcal{O}(s)$, for arbitrary $s$.

Lemma 2.16**.**

Let $\mathbb{P}(w_{1},\dots,w_{s})$ be any weighted projective space and let $l=\mathop{\rm lcm}\nolimits(w_{1},\dots,w_{s})$. The map from $\mathbb{P}(w_{1},\dots,w_{s})$ to a projective space given by all monomials of degree $k$ does not have a base locus if and only if $l|k$.

Proof.

If $w_{1}\nmid l$, then $[1:0:\dots:0]$ is a base point. If all $w_{i}|l$ then the monomials $x_{i}^{\frac{l}{w_{i}}}$ do not have a base locus. ∎

As there is some confusion in the literature, we stress that even if $l|k$ the map from the previous lemma does not have to be an embedding. In other words $\mathcal{O}(l)$, although always ample, may be not very ample.

Example 2.17**.**

The polytope $P((1,6,10,15),30)$ is not very ample. Equivalently, the map $\mathbb{P}(1,6,10,15)\to\mathbb{P}^{17}$ given by all monomials of weighted degree 30 is not an embedding. This can be checked with the software Normaliz [BIR*+*], or by studying the polytope.

This raises the following question: when is $\mathcal{R}_{d,k,m}$ an embedding of the weighted projective space $\mathbb{P}(1^{a_{1}},\dots,m^{a_{m}})$? We will prove that this happens if and only if $k$ is a multiple of all natural numbers between 1 and $m$. Equivalently, $\mathcal{O}(k)$ is a very ample line bundle if and only if $\mathop{\rm lcm}\nolimits(2,\dots,m)\mid k$. We will need the following technical lemma.

Lemma 2.18**.**

If $m\geq 7$ and $m\neq 10$, then there exist two distinct prime numbers strictly larger than $\frac{m}{2}$ and at most equal to $m$ such that their sum is not a power of two.

Proof.

If $m\leq 56$, we can easily find the required primes.

[TABLE]

Assume then $m\geq 57$. For $x\in\mathbb{N}$, let $\pi(x)$ be the number of prime numbers smaller or equal than $x$. In [RS62], the authors show that

[TABLE]

for every $x\geq 17$. Now,

[TABLE]

and the latter is at least 3 for $m\geq 57$, so there are at least three prime numbers $a<b<c$ between $\frac{m}{2}$ and $m$. If $b+c$ is not a power of 2, we are done. Otherwise we have $b+c=2^{n}$. Then

[TABLE]

so $a+b$ cannot be a power of 2. ∎

Theorem 2.19**.**

If $i\mid k$ for every $i\in\{1,\dots,m\}$, then $\mathcal{R}_{d,k,m}$ is projectively normal.

Proof.

For $m\leq 6$, normality can be explicitly checked with the software Polymake [AGH*+*17], so we assume $m\geq 7$. By Lemmas 2.14 and 2.15, we may assume that $k=\mathop{\rm lcm}\nolimits(1,\dots,m)$ and study the map

[TABLE]

defined by all monomials of weighted degree $k$. Let $P:=P((1,2,\ldots,m),k)$ be the associated polytope. For $s\in\mathbb{N}$, consider the dilation $sP$ of $P$. A lattice point of $sP$ is of the form $(a_{1},\dots,a_{m})\in\mathbb{N}^{m}$ with $a_{1}+2a_{2}+\ldots+ma_{m}=sk$. We can write it as a multiset

[TABLE]

By induction on $s$, we only need to prove that there exists a submultiset $S$ of $M$ whose entries sum up to $k$. Let us modify $M$ to $M^{\prime}$ in the following way. If there are $i,j\in M$ such that $i,j\leq\frac{m}{2}$, then discard $i$ and $j$ and add $i+j$. Notice that $i+j\leq m$. If $S^{\prime}$ is a submultiset of $M^{\prime}$ whose entries sum up to $k$, then either $S$ does not contain $i+j$, and therefore $S=S^{\prime}$ is a submultiset of $M$ as well, or $S^{\prime}$ contains $i+j$, and we define $S$ by replacing back $i+j$ with $i,j$. By iterating this argument, we can assume that there is at most one $i\in\{1,\dots,m\}$ such that $i\leq\frac{m}{2}$.

The resulting multiset $M^{\prime}$ is of the form $\left\{i_{1}^{b_{i_{1}}},\dots,i_{t}^{b_{i_{t}}}\right\}$. Let $k^{\prime}=\mathop{\rm lcm}\nolimits(i_{1},\dots,i_{t})$ and consider the map

[TABLE]

given by all the monomials of weighted degree $k^{\prime}$. Its image is associated to the polytope $P^{\prime}:=P((i_{1},i_{2},\ldots,i_{t}),k^{\prime})$. By construction, $k^{\prime}$ divides $k$ and $\dim P^{\prime}\leq t-1\leq\frac{m}{2}$. Observe that any point of $P^{\prime}$ has entries summing up to $k^{\prime}$. Since the entries of $M^{\prime}$ sum up to $sk$, $M^{\prime}$ is a point in $rP^{\prime}$ for $r=\frac{sk}{k^{\prime}}$. In fact, $M^{\prime}\in s\left(\frac{k}{k^{\prime}}P^{\prime}\right)$ and so, it is enough to prove that $\frac{k}{k^{\prime}}P^{\prime}$ is normal. This will imply that $M^{\prime}$ is a sum of lattice points of $\frac{k}{k^{\prime}}P^{\prime}$, and therefore it admits the required submultiset $S^{\prime}$.

If there exists a prime number $\frac{m}{2}\leq p\leq m$ such that $p\nmid k^{\prime}$, then $p\mid\frac{k}{k^{\prime}}$ and so $\frac{k}{k^{\prime}}\geq p\geq\frac{m}{2}\geq\dim P^{\prime}$. In this case, $\frac{k}{k^{\prime}}P^{\prime}$ is normal by [CLS11, Theorem 2.2.12]. This guarantees the existence of the desired $S^{\prime}$. Otherwise, assume $m\neq 10$. By Lemma 2.18, there exist two prime numbers $p_{1},p_{2}$ between $\frac{m}{2}$ and $m$ such that $p_{1}+p_{2}$ is not a power of $2$. Up to relabelling, we can assume that $b_{p_{1}}\leq b_{p_{2}}$.

Once more, we have to modify $M^{\prime}$ by making some replacements. For $b_{p_{1}}$ times, discard an entry $p_{1}$ and an entry $p_{2}$ and add an entry $p_{1}+p_{2}$. Denote by $M^{\prime\prime}$ the resulting multiset and by $P^{\prime\prime}$ the associated polytope, that satisfies $\dim P^{\prime\prime}\leq\frac{m}{2}$. As before, it is enough to find a submultiset $S^{\prime\prime}$ of $M^{\prime\prime}$ such that the entries of $S^{\prime\prime}$ sum up to $k$. By construction, the entries of $M^{\prime\prime}$ still sum up to $sk$ and their least common multiple $k^{\prime\prime}$ divides $k$. Indeed, the sum $p_{1}+p_{2}$ is even and not a power of two, thus all powers of prime numbers that divide it are smaller or equal to $m$. Notice that neither $p_{1}$, nor any of its multiples appear in $M^{\prime\prime}$. Therefore, $p_{1}\mid k$ and $p_{1}\nmid k^{\prime\prime}$, which implies that $p_{1}\mid\frac{k}{k^{\prime\prime}}$ and so $\frac{k}{k^{\prime\prime}}\geq p_{1}\geq\frac{m}{2}\geq\dim P^{\prime\prime}$. By [CLS11, Theorem 2.2.12], $\frac{k}{k^{\prime\prime}}P^{\prime\prime}$ is normal. This guarantees the existence of the desired $S^{\prime\prime}$.

The last case $m=10$ can be explicitly checked with the software Normaliz. ∎

Corollary 2.20**.**

The map $\mathbb{P}(1^{a_{1}},\dots,m^{a_{m}})\to\mathbb{P}^{N}$ given by all monomials of weighted degree $k$ is an embedding if and only if $i\mid k$ for every $i\in\{1,\dots,m\}$. If so, $\mathcal{R}_{d,k,m}\simeq\mathbb{P}(1^{a_{1}},\dots,m^{a_{m}})$ is embedded as a projectively normal variety.

Proof.

By Theroem 2.19, the polytope associated to $\mathcal{R}_{d,k,m}$ is normal and thus very ample. This implies that the map is an embedding and the image is projective normal. ∎

2.4 Dimension and Degree of the rough Veronese variety

In this section we provide formulas for the dimension and degree of $\mathcal{R}_{d,k,m}$. We work under the assumption that $k\geq m$, as otherwise some variables do not appear at all in the parametrization and hence we can easily reduce to this case. Recall that the normalized volume of a polytope $P\subset\mathbb{R}^{n}$, denoted by $\operatorname{vol}P$, is $n!$ times its Lebesgue measure.

Proposition 2.21**.**

Let $d,k,m\in\mathbb{N}$ with $d\geq 2$ and $k\geq m$. Let $W_{d,m}$ be the set of Lyndon words of length at most $m$ in the alphabet with $d$ letters.

The dimension of $\mathcal{R}_{d,k,m}$ is $\#(W_{d,m})-1$. 2. 2.

Let $l=\mathop{\rm lcm}\nolimits(1,2,\dots,m)$ and let $\Delta$ be the convex hull of the integral points

[TABLE]

where the number of occurrences of $\frac{l}{i}$ is the number of Lyndon words of length $i$. Then

[TABLE]

and

[TABLE]

Moreover, equality holds in (3) if and only if $l|k$.

The dimension of $\mathcal{R}_{d,k,m}$ was already computed in [AFS18, Remark 6.5] by proving that the fibers of the map $f_{d,k,m}$ have dimension 0. However, in our opinion it is interesting to have a different proof based on toric techniques.

Proof.

Let $a_{i}$ be the number of Lyndon words of length $i$. We know that $\mathcal{R}_{d,k,m}$ is the toric variety associated to the polytope

[TABLE]

Note that $\dim\mathcal{R}_{d,k,m}\leq\dim\mathop{\rm Lie}\nolimits^{m}(\mathbb{R}^{d})-1=\#(W_{d,m})-1$, as this is the dimension of the parameterizing weighted projective space. Thus, in order to prove the statement we only have to show that $P$ has maximal possible dimension. In fact, we prove that the lattice points of $P$ generate the lattice

[TABLE]

If $i\in\{1,\ldots,m\}$, $j\in\{1,\ldots,a_{i}\}$ and $(i,j)\neq(1,1)$, then there exists a lattice point $p\in P$ with $x_{i,j}=1$ and $x_{i^{\prime},j^{\prime}}=0$, unless $(i^{\prime},j^{\prime})=(i,j)$ or $(i^{\prime},j^{\prime})=(1,1)$. Take any $z\in L$. By adding and subtracting $p$ of the above type, we can reduce to the situation when $z$ has only the first coordinate nonzero. In such a case, the claim is obvious. 2. 2.

We note that $P$ is the convex hull of lattice points in $\frac{k}{l}\Delta$. Thus,

[TABLE]

Now, equality holds if and only if $P=\frac{k}{l}\Delta$. However, the latter is a lattice polytope if and only if $l|k$. Finally, note that $P((1^{a_{1}},\dots,m^{a_{m}}),k)$ is injected into $P((1^{a_{1}},\dots,m^{a_{m}}),k+1)$ simply by adding $(1,0,\dots,0)$. Hence

[TABLE]

which finishes the proof.∎

Many software-aided computations on the dimension, the degree and even the generators of the ideal of $\mathcal{R}_{d,k,m}$ are presented in [Gal18]. We now want to use Proposition 2.21 to deal with this numbers by using toric geometry.

Example 2.22**.**

Fix $d=m=2$. Since $W_{2,2}=\{1,2,12\}$, $\dim\mathcal{R}_{2,k,2}=2$. We now compute the degree for small values of $k$. Since $\mathcal{R}_{2,k,2}$ is a surface, we will deal with 2-dimensional polytopes. Here, $l=\mathop{\rm lcm}\nolimits(1,2)=2$ and we denote $P:=P((1^{2},2),k)$.

1. ($k=2$)

As in Notation 2.4, $R_{2,2}=\mathbb{C}[x^{2},xy,y^{2},a]$ and $P=\Delta$ is the convex hull of the points $(2,0,0),(0,2,0),(0,0,1)$. It is a triangle

[TABLE]

of normalized area 2, hence $\deg\mathcal{R}_{2,2,2}=2$. As **[Gal18, Example 15]** shows, $\mathcal{R}_{2,2,2}$ is the cone over a smooth conic in $\mathbb{P}^{3}$. 2. ($k=3$)

Since $R_{2,3}=\mathbb{C}[x^{3},x^{2}y,xy^{2},y^{3},xa,ya]$, $\frac{3}{2}\Delta=\mathop{\rm Conv}\nolimits\left\{(3,0,0),(0,3,0),(0,0,\frac{3}{2})\right\}$, while $P=\mathop{\rm Conv}\nolimits\{(3,0,0),(0,3,0),(1,0,1),(0,1,1)\}$.

[TABLE]

Therefore, $\deg\mathcal{R}_{2,3,2}=\operatorname{vol}P=4<\frac{9}{2}=\operatorname{vol}\frac{3}{2}\Delta$. 3. ($k=4$)

In this case, $R_{2,4}=\mathbb{C}[x^{4},x^{3}y,x^{2}y^{2},xy^{3},y^{4},x^{2}a,xya,y^{2}a,a^{2}]$, and so, $P=2\Delta=\mathop{\rm Conv}\nolimits\{(4,0,0),(0,4,0),(0,0,2)\}$ and $\deg\mathcal{R}_{2,4,2}=\operatorname{vol}P=8$.

[TABLE] 4. ($k=5$)

As in the case $k=3$, $P\subsetneq\frac{5}{2}\Delta$. More precisely, $\frac{5}{2}\Delta=\mathop{\rm Conv}\nolimits\left\{(5,0,0),(0,5,0),(0,0,\frac{5}{2})\right\}$, and $P=\mathop{\rm Conv}\nolimits\{(5,0,0),(0,5,0),(1,0,2),(0,1,2)\}$.

[TABLE]

Hence, $\deg\mathcal{R}_{2,5,2}=\operatorname{vol}P=12<\frac{25}{2}=\operatorname{vol}\frac{5}{2}\Delta$.

In this way, it is straightforward to see that, up to a linear isometry, the simplex $\frac{k}{2}\Delta$ is the triangle with vertices $(k,0,0),(0,k,0)$ and $\left(0,0,\frac{k}{2}\right)$. If $k$ is even, then they are lattice points, $P=\frac{k}{2}\Delta$, and

[TABLE]

If $k$ is odd, then $P$ is the trapezium with vertices $(k,0,0),(0,k,0),\left(1,0,\frac{k-1}{2}\right)$ and $\left(0,1,\frac{k-1}{2}\right)$, implying that

[TABLE]

3 Axis-parallel paths

Besides $\mathcal{R}_{d,k,m}$, the universal variety contains other interesting subvarieties. One of them is the signature variety $\mathcal{L}_{d,k,m}$ of piecewise linear paths. Up to translation, a piecewise linear path $X$ with $m$ steps is the concatenation of $m$ linear paths, each represented by a vector $v_{i}$. This decomposition is unique, provided that $v_{i+1}$ is not a multiple of $v_{i}$ for any $i$. In this section, we study the subfamily of axis-parallel paths.

Definition 3.1**.**

Let $\{e_{1},\dots,e_{d}\}$ be the standard basis for $\mathbb{R}^{d}$. A piecewise linear path $X=v_{1}\sqcup\ldots\sqcup v_{m}$ is an axis-parallel path (or simply an axis path) if there are $a_{1},\dots,a_{m}\in\mathbb{R}$ such that $v_{i}=a_{i}e_{\nu_{i}}$ for every $i$, where $\nu_{i}\in\{1,\dots,d\}$.

In other words, each step is a multiple of a basis vector. Therefore, an axis-parallel path is characterized by two sequences. One of them is the sequence $\nu=(\nu_{1},\dots,\nu_{m})$, called shape of $X$, that stores in each $\nu_{i}\in\{1,2,\dots,d\}$ the direction of the $i$-th step. The other sequence, $a=(a_{1},\dots,a_{m})\in\mathbb{R}^{m}$, stores the length of each step and we call it the sequence of lengths. Notice that $\ell(\nu)=\ell(a)=m$. When we study an axis-parallel path $X\colon[0,1]\to\mathbb{R}^{d}$, we may assume that the image is not contained in any hyperplane, that is, it is nondegenerate. This means that $\{\nu_{1},\dots,\nu_{m}\}=\{1,\dots,d\}$.

The $k$-th signature of an axis-parallel path $X$ can be computed combinatorially in a very nice way. Recall that a partition of a set $S$ is a collection of subsets (called blocks) such that their union is $S$. Now each sequence $\nu$ induces a partition $\pi_{\nu}=\{\pi_{1}|\pi_{2}|\dots|\pi_{d}\}$ of the set $\{1,\dots,m\}$, defined by $(\pi_{\nu})_{i}=\{j\in\{1,\dots,m\}\mid\nu_{j}=i\}$. For instance, if $\nu=(1,2,1,3,3,1)$ then $\pi_{\nu}=\{1,3,6|2|4,5\}$. We will write $\pi$ instead of $\pi_{\nu}$ when there are no ambiguities. We can now introduce the main character of this section.

Definition 3.2**.**

Fix $\nu=(\nu_{1},\dots,\nu_{m})$ and $k\in\mathbb{N}$. Let $d=\max(\nu_{1},\ldots,\nu_{m})$. For an axis parallel path $X=a_{1}e_{\nu_{1}}\sqcup\ldots\sqcup a_{m}e_{\nu_{m}}$, let $g_{\nu,k}(X):=\sigma^{(k)}(X)$ be its $k$-th signature. As we did in Definition 2.3, we pass to the complex projective space and we define the axis paths variety $\mathcal{A}_{\nu,k}$ to be closure of the image of the composition

[TABLE]

Instead of $g_{\nu,k}$, we can also consider

[TABLE]

by sending $a\mapsto(g_{\nu,1}(a),\dots,g_{\nu,k}(a))$. In this case we denote by $\mathcal{A}_{\nu,\leq k}\subset\mathbb{C}^{d}\times(\mathbb{C}^{d})^{\otimes 2}\times\ldots\times(\mathbb{C}^{d})^{\otimes k}$ the closure of the complexification of the image of $G_{\nu,k}$.

There is a nice way to write down the polynomials defining the map $g_{\nu,k}$. The following result is a consequence of [AFS18, Corollary 5.3].

Lemma 3.3**.**

Let $X$ be the axis-parallel path of shape $\nu$ and sequence of lengths $a$. Then the $(i_{1}\dots i_{k})$-th entry of the $k$-th signature is

[TABLE]

where we sum over all the non-decreasing sequences $(j_{1},j_{2},\dots,j_{k})$ such that $j_{l}\in\pi_{i_{l}}$ for $l\in\{1,\dots,k\}$, and $s_{l}$ counts the number of times that $l$ appears in $(j_{1},j_{2},\dots,j_{k})$.

Lemma 3.3 allows us to find the $k$-th signature of an axis-parallel path and to explicitly write the map $g_{\nu,k}$. We implemented a Macaulay2 code to compute the ideal of $\mathcal{A}_{\nu,k}$. This code can be found in [CGM].

Example 3.4**.**

Let $\sigma$ be the 4-th signature of an axis path of shape $\nu=(1,2,1,3,2,3,1,4)$ and sequence of lengths $a=(a_{1},\dots,a_{8})$. Then $\pi_{\nu}=\{1,3,7|2,5|4,6|8\}$. By Lemma 3.3,

[TABLE]

Remark 3.5**.**

As any signature variety, the axis paths variety is contained in the universal variety. Namely, $\mathcal{A}_{\nu,k}\subset\mathcal{U}_{d,k}$ for every $d\geq\max(\nu_{1},\ldots,\nu_{m})$. It is known that $\mathcal{L}_{d,k,m}$ coincides with $\mathcal{U}_{d,k}$ for $m$ large enough, because every piecewise smooth path can be approximated by using piecewise linear paths. Since every piecewise linear path can be approximated by an axis path, we deduce that for every $d$ and $k$, there exists $\nu$ such that $\max(\nu_{i})=d$ and $\mathcal{A}_{\nu,k}=\mathcal{U}_{d,k}$.

It is worth to write down what Chen’s identity means for the axis paths signature variety.

Proposition 3.6**.**

Consider two sequences $\nu_{1}$ and $\nu_{2}$. If $X_{1}$ and $X_{2}$ are axis-parallel paths of shape $\nu_{1}$ and $\nu_{2}$ respectively, then

[TABLE]

In particular, $\mathcal{A}_{\nu_{1}\nu_{2},\leq k}$ is a projection of the Segre product $\mathcal{A}_{\nu_{1},\leq k}\times\mathcal{A}_{\nu_{2},\leq k}$.

In Section 2 we saw how important it is for a variety to be toric, so it is natural to ask whether $\mathcal{A}_{\nu,k}$ enjoys such property.

3.1 Toricness of $\mathcal{A}_{\nu,k}$

It is not difficult to find instances when the axis paths variety is toric. As an example we can take $\mathcal{A}_{\nu,1}=\mathbb{R}^{d}$, or consider $\nu=(1,2,\dots,d)$ and obtain the Veronese variety $\mathcal{V}_{d,k}=\mathcal{A}_{\nu,k}$. A less trivial example is given by Remark 3.5. Thanks to Lemma 3.3, we are able to perform efficient computations and check that many other occurrences of $\mathcal{A}_{\nu,k}$ are toric, and even find their polytopes.

Example 3.7**.**

In [CGM] we compute the ideal of $\mathcal{A}_{(1,2,1),3}$ and we check that it is toric. Further computations on the ideal allow us to determine its polytope $P$. Indeed, $\mathcal{A}_{(1,2,1),3}$ is a degree 6 surface in $\mathbb{P}^{6}$, so $P$ is a two-dimensional lattice polytope of normalized area 6 that contains exactly 7 lattice points. By checking the Hilbert and Erhart polynomials and the Betti table, we see that there are only two possible polytopes.

\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\circ

\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\circ

By looking at the intersections of the tangent spaces at singular points with the variety, we see that the correct one is the one on the right.

In [Gal18] the second author presents a change of coordinates, based on the exponential description of the tensor signature, for which the universal variety $\mathcal{U}_{d,k}$ is the image of a monomial map, and therefore it is toric. However, such change of coordinates fails to prove that every $\mathcal{A}_{\nu,k}$ is toric as well. For instance, it does not turn the ideal of $\mathcal{A}_{(1,2,1),3}$ into a binomial ideal. One could still hope to find another change of coordinates that makes both the ideals of $\mathcal{A}_{\nu,k}$ and $\mathcal{U}_{d,k}$ binomial. Unfortunately, this is not true.

Example 3.8**.**

The polytopes $P$ and $Q$ of $\mathcal{U}_{3,3}$ and $\mathcal{A}_{(1,2,1),3}$ are

\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\circ

There are two cases how $\mathcal{A}_{(1,2,1),3}$ could be a toric subvariety of $\mathcal{U}_{3,3}$. Either it is a toric divisor or the inclusion is a toric morphism. A careful technical analysis shows that no change of coordinates in $\mathbb{P}^{7}$ can simultaneously make the ideals of both $\mathcal{A}_{(1,2,1),3}$ and $\mathcal{U}_{3,3}$ binomial.

Despite previous Example, the axis paths variety turns out to be toric in many cases.

Proposition 3.9**.**

Let $d=\max(\nu_{1},\ldots,\nu_{m})$. Then

$\mathcal{A}_{\nu,3}$* is toric for every $d\leq 2$.* 2. 2.

$\mathcal{A}_{\nu,2}$* is toric for every $d\leq 3$.* 3. 3.

$\mathcal{A}_{(1,2,\dots,d,j),2}$* is toric for every $j\in\{1,\dots,d-1\}$.*

Proof.

It is not difficult to check all possible cases for the first two items. The computation can be found in [CGM]. For the third item, we have $m=d+1$. Let us apply the change of coordinates

[TABLE]

on the domain $\mathbb{R}^{d+1}$. By Lemma 3.3, we know that the entries of $\sigma^{(2)}(X)$ are

[TABLE]

Finally, if we perform the change of coordinates $\sigma^{(2)}(X)_{jd}\mapsto\sigma^{(2)}(X)_{jd}+\sigma^{(2)}(X)_{j,d+1}$, the map becomes monomial. ∎

There are further cases in which we can prove that the axis paths variety is toric.

Lemma 3.10**.**

If the ideal of $\mathcal{A}_{\nu,\leq k}$ is binomial after a linear change of coordinates, then the same holds for the ideal of $\mathcal{A}_{\nu(d+1),k}$.

Proof.

Let $X$ be an axis path of shape $\nu$ and sequence of length $(a_{1},\dots,a_{m})$. Since $d+1$ does not appear in $\nu$, in the set partition $\pi_{\nu(d+1)}$ the $(d+1)$-st block only contains the entry $a_{m+1}$. Therefore, by Lemma 3.3,

[TABLE]

Hence, $\mathcal{A}_{\nu(d+1),k}$, up to diagonal change of coordinates, is equal to the projectivization of $\mathcal{A}_{\nu,\leq k}$. ∎

We finish this section with the following natural question.

Question**.**

Is every variety $\mathcal{A}_{\nu,k}$ toric?

Our results provide a positive answer in all relatively small cases. However, since we found no general technique, we expect that the answer may be negative. The main obstacle to provide a nontoric example is that a first potential candidate is already too large to be dealt with, either with ad hoc geometric arguments or general computational methods [KMM17].

3.2 Dimension of $\mathcal{A}_{\nu,k}$

As we stressed in the introduction, a fundamental problem when we deal with signatures is to determine whether it is possible to recover the path from the signature. The best case scenario is injectivity, when we can uniquely reconstruct the path. In general this is too much to hope, so we focus on a weaker property, and we want to know when $g_{\nu,k}$ is generically finite. Therefore we are interested in the dimension of the fibers, or equivalently in $\dim\mathcal{A}_{\nu,k}$. Since $\mathcal{A}_{\nu,k}\subset\mathcal{U}_{d,k}$ is the image of $\mathbb{R}^{\ell(\nu)}$ under a polynomial map, we have

[TABLE]

In general, the inequality can be strict. For instance, the dimension of $\mathcal{A}_{(1,2,1,2,3),3}$ is 4, while $\ell(\nu)=5$ and $\dim(\mathcal{U}_{3,3})=7$. In order to better study the dimension of our axis-parallel signature variety, we introduce the following definition.

Definition 3.11**.**

For $d,k\in\mathbb{N}$, we define the set

[TABLE]

We say that $\mathcal{A}_{\nu,k}$ is defective if $\nu\in N_{d,k}$ for $d\geq\max(\nu)$. Otherwise, we say that $\mathcal{A}_{\nu,k}$ has the expected dimension.

It may seem that we made a choice in using $G_{\nu,k}$ instead of $g_{\nu,k}$ in the previous definition. However, the $k$-th signature of a generic piecewise smooth path $X$ defines all the previous signatures up to finitely many choices (see [AFS18, Section 6]). Therefore,

[TABLE]

For the same reason, $N_{d,k+1}\subset N_{d,k}$ for every $k$. Each $N_{d,k}$ is a language, i.e. a set of words. It would be very interesting to completely characterise it - cf. Conjecture 3.15. First, we prove that $N_{d,k}$ is absorbing with respect to concatenation.

Lemma 3.12**.**

Let $\nu_{1},\nu_{2}$ be two shapes, let $\nu_{1}\nu_{2}$ be their concatenation and let $d=\max(\nu_{1}\nu_{2})$. If $\nu_{1}\in N_{d,k}$, then both $\nu_{1}\nu_{2}$ and $\nu_{2}\nu_{1}$ belong to $N_{d,k}$.

Proof.

Take a general point of $\mathcal{A}_{\nu_{1}\nu_{2},\leq k}$. It is of the form $\sigma^{\leq k}(X)$ for some general axis-parallel path $X$. Since $X$ has shape $\nu_{1}\nu_{2}$, we can write it as a concatenation $X_{1}X_{2}$, for general paths $X_{1}$ of shape $\nu_{1}$ and $X_{2}$ of shape $\nu_{2}$. By hypothesis, $\nu_{1}\in N_{d,k}$, and so there exist infinitely many paths $Y$ of shape $\nu_{1}$ such that $g_{\nu_{1},a}(Y)=g_{\nu_{1},a}(X_{1})$ for every $a\leq k$. By Proposition 3.6,

[TABLE]

hence the fiber containing $X$ is not finite and therefore $\nu_{1}\nu_{2}\in N_{d,k}$. In the same way it is possible to prove that $\nu_{2}\nu_{1}\in N_{d,k}$. ∎

On the other hand, if $\mathcal{A}_{\nu,k}$ is not defective, then we can add a new letter at any point in $\nu$ and the resulting variety is still of expected dimension.

Lemma 3.13**.**

Let $\nu\notin N_{d,k}$ and let $d=\max(\nu)$. If we write $\nu=\nu_{1}\nu_{2}$ and we take a new letter $l>d$, then $\nu_{1}\cdot l\cdot\nu_{2}\notin N_{l,k}$.

Proof.

Let us pick a general element in the affine cone over $\mathcal{A}_{\nu_{1}\cdot l\cdot\nu_{2},k}$. First we know that, up to finite number of choices, we can identify the parameter $a$ associated to $l$, as $\sigma_{(l^{k})}=\frac{a^{k}}{k!}$. We may also identify the other parameters as the signatures indexed by numbers from one to $d$ are the same for $\nu$ and $\nu_{1}\cdot l\cdot\nu_{2}$. ∎

We noticed several times that the universal variety can be obtained as $\mathcal{A}_{\nu,k}$ for sufficiently long $\nu$. Now we want to prove that this can be done in an efficient way, namely by using a sequence with as many entries as $\dim\mathcal{U}_{d,k}$.

Lemma 3.14**.**

For every $d$ and $k$, there is a shape $\nu$ such that $\max(\nu)=d$, $\mathcal{A}_{\nu,k}=\mathcal{U}_{d,k}$ and $\ell(\nu)=\dim\mathcal{U}_{d,k}$.

Proof.

By Remark 3.5, there is a $\nu^{\prime}$ satisfying the first two properties. We build $\nu$ as a subsequence of $\nu^{\prime}$ by deleting those entries that do not increase the dimension of the variety in the following way. If $\nu^{\prime}$ satisfies the last requirement too, we are done. Suppose then that $\nu^{\prime}=(\nu_{1},\dots,\nu_{s})$ for $s>\dim\mathcal{U}_{d,k}$. By construction, there is an index $i\in\{1,\dots,s-1\}$ such that $\mathcal{A}_{(\nu_{1},\dots,\nu_{i}),k}=\mathcal{A}_{(\nu_{1},\dots,\nu_{i+1}),k}$. If $i=s-1$, take $\nu^{\prime}:=(\nu_{1},\dots,\nu_{s-1})$ and proceed by induction on the length $\nu^{\prime}$. Otherwise, let $\nu:=(\nu_{1},\dots,\widehat{\nu_{i+1}},\dots,\nu_{s})$, where the entry with the hat is omitted. Consider the sequences

[TABLE]

so that $\nu^{\prime}=\alpha^{\prime}\beta$ and $\nu=\alpha\beta$. By Proposition 3.6, the map

[TABLE]

factors as

[TABLE]

We also have another map

[TABLE]

Since $\mathcal{A}_{\alpha,\leq k}=\mathcal{A}_{\alpha^{\prime},\leq k}$, we conclude that $\mathcal{A}_{\nu,\leq k}=\mathcal{A}_{\nu^{\prime},\leq k}$. We continue the process until there is no such index $i$. The resulting subsequence of $\nu^{\prime}$ satisfies the requirements of the statement. ∎

As an easy consequence, we notice that if $d=2$ there is only one possible shape $(1,2,1,2,\dots)$. Since it is unique, it satisfies the properties of Lemma 3.14 and therefore $\mathcal{A}_{\nu,k}$ has always the expected dimension.

Now, we look back to our first defective example. The reason why $\mathcal{A}_{(1,2,1,2,3),3}$ is defective is that there is a subsequence $(1,2,1,2)$ such that $\mathcal{A}_{(1,2,1,2),3}=\mathcal{U}_{2,3}$ but $\dim(\mathcal{U}_{2,3})=3$. This means that we are filling a 3-dimensional universal variety with a sequence of length four. We conjecture that this behavior is the only obstruction to having the expected dimension.

Conjecture 3.15**.**

Let $\nu=(\nu_{1},\dots,\nu_{m})$ be a sequence with $d=\max(\nu)$. Then $\mathcal{A}_{\nu,k}$ is defective if and only if there is a subsequence $\nu^{\prime}=(\nu_{i},\dots,\nu_{i+r})$ of $\nu$ with $d^{\prime}=\max(\nu^{\prime})$, such that $\mathcal{A}_{\nu^{\prime},k}=\mathcal{U}_{d^{\prime},k}$ but $r+1>\dim(\mathcal{U}_{d^{\prime},k})$.

3.3 Determinant of axis-parallel signatures

Let us start with an observation about the entries of the $k$-th signature for a special family of axis-parallel paths.

Lemma 3.16**.**

Let $k\in\mathbb{N}$ and let $X$ be an axis-parallel path with shape $\nu$ and sequence of length $(a_{1},\dots,a_{m})$. If an entry $\nu_{i}$ of $\nu$ appears only once in the $j$-th block of $\pi$, then $a_{i}$ divides all the entries of each $j$-th slice of $\sigma^{(k)}(X)$. If moreover $k=2$, then $a_{i}^{2}\mid\det(\sigma^{(2)}(X))$.

Proof.

If we look at one of the $j$-th slices, we are fixing one of the indices of $\sigma^{(k)}(X)$ to be $j$. By Lemma 3.3, every monomial of the sum is a multiple of $a_{i}$. When $k=2$, we are dealing with a square matrix in which both the $j$-th row and the $j$-th column of $\sigma^{(2)}(X)$ are multiples of $a_{i}$. In order to conclude that $a_{i}^{2}\mid\det(\sigma^{(2)}(X))$, it is enough to check that $a_{i}^{2}$ divides the diagonal entry $\sigma^{(2)}(X)_{jj}$, and this follows from Lemma 3.3. ∎

This is just a special case of a much more general phenomenon, corroborated by many experiments with Sage [The18] and the code included in [CGM]. The determinant of the second signature of an axis-parallel path is always the square of a polynomial in $a_{1},\dots,a_{m}$. This subsection is devoted to explaining this result and its consequences. In order to correctly state it, we need some definitions.

Definition 3.17**.**

For any shape $\nu=(\nu_{1},\dots,\nu_{m})$ with $d=\max\nu$, we say that a subsequence $\mu=(\nu_{i_{1}},\dots,\nu_{i_{d}})$ is a good subshape if $i_{1}<\dots<i_{d}$ and $\{\nu_{i_{1}},\dots,\nu_{i_{d}}\}=\{1,\dots,d\}$. Moreover, we define the sign of a good subshape, $\operatorname{sgn}\mu$, as the sign of the permutation $(\nu_{i_{1}},\dots,\nu_{i_{d}})\in S_{d}$.

Without loss of generality we assumed that $\{1,\dots,d\}=\{\nu_{1},\dots,\nu_{m}\}$, so good subshapes always exist.

Definition 3.18**.**

Let $X$ be an axis path of shape $\nu$ and let $a_{i}$ be the parameter associated to $\nu_{i}$. We define the degree $d$ homogeneus polynomial $P(a)\in\mathbb{C}[a_{1},\dots,a_{m}]$ by

[TABLE]

where the sum is taken over all good subshapes $\mu$ of $\nu$. Moreover, we denote by $\det_{2}\nu\in\mathbb{C}[a_{1},\dots,a_{m}]$ the determinant $\det\left(\sigma^{(2)}(X)\right)$. By Lemma 3.3, it is homogeneous of degree $2d$.

Theorem 3.19**.**

With the notation from Definition 3.18, we have

[TABLE]

The proof we found is quite technical and we present it in Appendix A. However, Theorem 3.19 has several interesting consequences. Since we can write $\mathcal{U}_{d,2}$ as an axis paths variety, we now know that the determinant of the signature matrix of any path is a square. In particular, if $X$ is any path, then $\det\sigma^{(2)}(X)\geq 0$. It is even more interesting to think in terms of the shuffle identity.

Example 3.20**.**

Let $X\colon[0,1]\to\mathbb{R}^{2}$ be any path. By Lemma 1.5,

[TABLE]

In this case it was not too difficult to realize that $4(11\shuffle 22-12\shuffle 21)=(12-21)^{\shuffle 2}$ and therefore that $\det(\sigma^{(2)}(X))$ is a square. Note that $12-21$ is twice the Lévy area defined by the path. This is a general behavior.

Definition 3.21**.**

Let $S_{d}$ be the symmetric group in $d$ elements. Define

[TABLE]

It is the sum of $d!$ words of length $d$, so it is a degree $d$ element of the shuffle algebra.

On one hand, this has a geometric meaning: as illustrated in [DR18, Section 3.2], $\mathop{\rm inv}\nolimits_{d}$ has an interpretation in terms of the signed volume of the convex hull of the path. On the other hand, there is a relation with signatures.

Lemma 3.22**.**

Let $a=(a_{1},\dots,a_{m})\in\mathbb{R}^{m}$ and let $X\colon[0,1]\to\mathbb{R}^{d}$ be the axis path corresponding to the sequence of lengths $a$ and shape $\nu$. Then $P(a)=\langle\sigma(X),\mathop{\rm inv}\nolimits_{d}\rangle$.

Proof.

By Lemma 3.3, if $i_{1},\dots,i_{d}$ are pairwise disjoint then $\langle\sigma(X),i_{1}\dots i_{d}\rangle$ equals the sum of products $\prod_{l=1}^{d}a_{j_{l}}$ where $\nu_{j_{l}}=i_{l}$ and $j_{1}<\dots<j_{d}$. In particular, it is a subsum in the definition of $P(a)$ corresponding to the good subshapes associated to the permutation $(i_{1},\dots,i_{d})$. Summing up over all possible permutations $(i_{1},\dots,i_{d})$ with signs we obtain the statement of the lemma. ∎

It follows that Theorem 3.19 has an important consequence on the shuffle algebra.

Corollary 3.23**.**

Consider the matrix

[TABLE]

with coefficients in the shuffle algebra $(T(\mathbb{R}^{d}),\shuffle,e)$. Let $\det_{\shuffle}(A)$ be its determinant, where the product is the shuffle. Then $2^{d}\det_{\shuffle}(A)=(\mathop{\rm inv}\nolimits_{d})^{\shuffle 2}$.

Proof.

It is enough to prove that $\langle T,(\mathop{\rm inv}\nolimits_{d})^{\shuffle 2}\rangle=2^{d}\langle T,\det_{\shuffle}(A)\rangle$ for every $T\in T((\mathbb{R}^{d}))$. Since the linear span of $\mathcal{G}(\mathbb{R}^{d})$ is the whole $T((\mathbb{R}^{d}))$, by linearity we just have to show that such equality holds when $T\in\mathcal{G}(\mathbb{R}^{d})$. Let us pick then $T\in\mathcal{G}(\mathbb{R}^{d})$. Then there is an axis path $X$, corresponding to the sequence of lengths $a\in\mathbb{R}^{m}$, such that $\langle T,(\mathop{\rm inv}\nolimits_{d})^{\shuffle 2}\rangle=\langle\sigma(X),(\mathop{\rm inv}\nolimits_{d})^{\shuffle 2}\rangle$ and $\langle T,\det_{\shuffle}(A)\rangle=\langle\sigma(X),\det_{\shuffle}(A)\rangle$. By Theorem 3.19 and Lemma 3.22,

[TABLE]

Even though the techniques we use are based on the combinatorics of $\mathcal{A}_{\nu,k}$, Corollary 3.23 has nothing to do with axis paths, nor with signatures at all. It would be very interesting to find a generalization for $k>2$. In our opinion, this can give new insight on the properties of the shuffle algebra.

Appendix A Proof of Theorem 3.19

For this proof we fix an axis path $X$ with shape $\nu\in\mathbb{R}^{m}$ and sequence of lengths $a\in\mathbb{R}^{m}$. The first step will be to reduce the problem to the case in which every entry of $\nu$ appears at most twice. Let us start with an observation.

Remark A.1**.**

The symmetric group $S_{d}$ acts on $\mathbb{R}^{d}$ by permuting the basis elements. In the same way, it acts on the sequence $\nu$, on the path $X$ and thus on the polynomials $P$ and $\det_{2}\nu$. The determinant is invariant, while the sign of $P$ changes according to the sign of the permutation. However, $P^{2}$ is invariant. This means that we are allowed to relabel the entries of $\nu$. In other words, we can always pick $\sigma\in S_{d}$ and replace $(\nu_{1},\dots,\nu_{m})$ with $(\sigma(\nu_{1}),\dots,\sigma(\nu_{m}))$.

In order to prove that two polynomials are equal, it is enough to prove that they have the same coefficient on each monomial.

Notation A.2**.**

For a polynomial $Q$ and a monomial $M$ let $Q_{|M}$ be the coefficient of $Q$ corresponding to $M$.

Since both $P^{2}$ and $\det_{2}\nu$ are homogeneous of degree $2d$, we only have to take care of degree $2d$ monomials.

Lemma A.3**.**

Suppose that there are three (not necessarily distinct) indices $i,j,l\in\{1,\dots,m\}$ such that $\nu_{i}=\nu_{j}=\nu_{l}$.

Let $M\in\mathbb{C}[a_{1},\dots,a_{m}]_{2d}$ be such a monomial that $a_{i}a_{j}a_{l}\mid M$. Then $P^{2}_{|M}=(\det_{2}\nu)_{|M}=0$. 2. 2.

Let $\nu^{\prime}$ be the sequence obtained from $\nu$ by removing $\nu_{l}$ and let $P^{\prime}$ be the associated polynomial. If $M\in\mathbb{C}[a_{1},\dots,\hat{a_{l}},\dots,a_{m}]_{2d}$ is a monomial, then $P^{\prime}_{|M}=P_{|M}$ and $\det_{2}\nu_{|M}=\det_{2}\nu^{\prime}_{|M}$.

Proof.

Thanks to remark A.1, we may assume $\nu_{i}=\nu_{j}=\nu_{l}=1$. By Lemma 3.3, the variables $a_{i},a_{j},a_{l}$ only appear in the first row and in the first column of the signature matrix. Furthermore, among the monomials of the diagonal entry $\sigma(X)_{11}$ there are degree at most 2 monomials in the variables $a_{i},a_{j},a_{l}$, while away from the diagonal all monomials contain only one of these variables, with exponent 1. Therefore $a_{i}a_{j}a_{l}$ does not divide any monomial appearing in $\det_{2}\nu$. On the other hand, every good subshape $\mu$ of $\nu$ contains the entry 1 exactly once, so each monomial of $P$ contains at most one among $a_{i},a_{j},a_{l}$, and with exponent 1. In $P^{2}$ there can be monomials containing the product of two among $a_{i},a_{j},a_{l}$, but not containing all three of them. The second statement follows from the definitions of $P$ and $\det_{2}\nu$. ∎

Thanks to Lemma A.3, from now on we can restrict our attention to sequences $\nu$ with no triple entries. Now we want to take care of the case in which an entry appears only once. We find it useful to focus on a particular monomial.

Definition A.4**.**

Assume that $\nu$ has no triple entries. If $\mu=(\nu_{i_{1}},\dots,\nu_{i_{\ell}})$ is a subsequence of $\nu$, we set

[TABLE]

and we define the monomial

[TABLE]

Since we are assuming that $\nu$ has no triple entries, $\deg(M_{\nu})=2d$.

Lemma A.5**.**

If a monomial $M$ appears in either $P^{2}$ or $\det_{2}\nu$ and it is divisible by all variables $a_{1},\dots,a_{m}$, then $M=M_{\nu}$.

Proof.

Let $M$ be such a monomial. By Lemma A.3, $M\mid M_{\nu}$. Indeed, if $\nu_{i}$ appears in $\nu$ at most once, then $a_{i}^{3}\nmid M$. If $\nu_{i}=\nu_{j}$ appears exactly twice then $a_{i}a_{j}\mid M$, but $a_{i}^{2},a_{j}^{2}\nmid M$. As the degrees are the same, equality follows. ∎

The next results will explain what happens to the coefficient of $M_{\nu}$ when $\nu$ has a non-repeated entry.

Lemma A.6**.**

Assume $\nu$ contains at least one non-repeated entry. Suppose that the last one occurs in $\nu_{m-l}$. Let $\nu_{0}$ be the sequence obtained from $\nu$ by discarding $\nu_{m-l}$ and let $P_{0}$ be the associated polynomial. Then

[TABLE]

Proof.

By Remark A.1, we can assume that $\nu_{m-l}=d$. Since $M_{\nu}$ is a monomial of $P^{2}$, to compute $P^{2}_{|M_{\nu}}$ we have to consider the contribution of all pairs $(\alpha,\beta)$ of good subshapes of $\nu$. Let $(\alpha,\beta)$ be a pair contributing to $M_{\nu}$. The variable of any non-repeated entry - including $d$ - has to appear in both $\alpha$ and $\beta$. On the other hand, the variable of an entry that appears twice appears once in $\alpha$ and once in $\beta$. Therefore there is a bijection between pairs $(\alpha,\beta)$ of good subshapes of $\nu$ contributing to $M_{\nu}$ and pairs $(\alpha_{0},\beta_{0})$ of good subshapes of $\nu_{0}$ contributing to $M_{\nu_{0}}$, where $\alpha_{0}$ is the sequence obtained from $\alpha$ by removing $d$, and $\beta_{0}$ is defined in a similar way from $\beta$. If we set now $a$ (resp. $b$) to be the number of entries of $\alpha$ (resp $\beta$) after the discarded one, then

[TABLE]

Before we prove the analogous statement for the polynomial $\det_{2}\nu$, we introduce a combinatorial interpretation of the coefficient ${\det}_{2}(\nu)_{|M_{\nu}}$. By Laplace expansion, the contributions to ${\det}_{2}(\nu)_{|M_{\nu}}$ come from directed graphs (with possible loops) with vertices corresponding to symbols in $\nu$ and:

• •

if a symbol $i$ appears twice in $\nu$ then precisely one such vertex is outgoing and one is incoming;

• •

if a symbol $i$ appears once in $\nu$ then it has degree two and has one incoming and one outgoing edge - this is the only case where a vertex can be a loop;

• •

all edges are from left to right.

Each such graph contributes $\pm\frac{1}{2^{m}}$ where $m$ is the number of loops and the sign is the sign of the corresponding permutation.

Example A.7**.**

Let $\nu=(1,2,1)$. We have two possible graphs. One with edges $(1,2)$ and $(2,1)$. It encodes the transposition $(1,2)$ and contributes with $-1$. The other one has an edge $(1,1)$ and a loop over two. It encodes the identity permutation and contributes $\frac{1}{2}$.

Lemma A.8**.**

With the same hypothesis as in Lemma A.6,

[TABLE]

Proof.

We look at the possible cases for the position of $d$.

If $d$ is the last symbol then the $d$-th column of the second signature matrix has only one nonzero entry, which equals $\frac{1}{2}$ times the square of the variable associated to $d$. Applying Laplace expansion we obtain the formula in this case.

We now assume that $d$ is the last but one entry. The contributing graphs are of two types.

If $d$ is a loop then (by forgetting the loop) we obtain graphs on $\nu_{0}$. Hence, the contribution of those graphs is $\frac{1}{2}({\det}_{2}(\nu_{0}))_{|M_{\nu_{0}}}$.

If $d$ is not a loop, it must have an incoming edge, say $(a,d)$ and an outgoing edge $(d,b)$, where $b$ must be the last symbol in $\nu$. We may replace this by an edge $(a,b)$. In the cycle presentation of the permutation we removed $d$ from one cycle, i.e. changed the sign of the permutation. As we did not change the number of loops the contribution equals $-({\det}_{2}(\nu_{0}))_{|M_{\nu_{0}}}$. Summing the two contributions we obtain the result in this case.

We may now assume that $d$ is the last letter in $\nu$ that appears once and further there are at least two symbols after it, i.e. $\nu=\dots dab\dots$. Let $\nu^{\prime}=\dots abd$ be $\nu$ with $d$ moved two places forward. We will prove that $({\det}_{2}(\nu))_{|M_{\nu}}=({\det}_{2}(\nu^{\prime}))_{|M_{\nu^{\prime}}}$. By induction this will finish the proof of the lemma.

We may identify graphs contributing to $({\det}_{2}(\nu))_{|M_{\nu}}$ with those contributing to $({\det}_{2}(\nu^{\prime}))_{|M_{\nu^{\prime}}}$ with the exception of graphs for which there is an edge from $d$ either to $a$ or $b$. Further, if there is an edge $(d,a)$ and the considered $b$ is an outgoing vertex, the encoded permutation is $x\rightarrow d\rightarrow a$, $b\rightarrow c$ for some $x$ and $c$. We may associate to it a graph on $\nu^{\prime}$ with edges $(x,a),(b,d),(d,c)$. This does not change the sign of permutation. We proceed in the same way if there is an edge $(d,b)$ and $a$ is outgoing. The remaining graphs are those where there is an edge from $d$ to $a$ or $b$ and the other vertex is incoming. We will prove that the sum of contributions of such graphs equals zero.

Consider a graph with edges $(d,a),(c,b)$ for some $c$. We associate to it a graph with edges $(c,a),(d,b)$. In the cycle decomposition of the permutation this operation either joins two cycles or decomposes one cycle into two, i.e. changes the sign of the permutation. In particular, the contribution of each pair equals zero.

We also see that we obtain all graphs on $\nu^{\prime}$, apart from those for which there is an edge from $a$ or $b$ to $d$ and the other vertex is outgoing. Just as above one can show that the contribution of such graphs equals zero. ∎

We can now reduce to the case in which every entry of $\nu$ appears exactly twice.

Corollary A.9**.**

Assume Theorem 3.19 holds for sequences in which every entry appears exactly twice. Then it holds for every sequence.

Proof.

We compare the coefficients of monomials in both polynomials. If a monomial is different from $M_{\nu}$, then it does not contain one of the variables. In this case Lemma A.5 allows us to consider a subsequence of $\nu$ and conclude by induction on $\ell(\nu)$. Consider then $M_{\nu}$. By repeatedly applying Lemmas A.6 and A.8, we can discard all non-repeated entries and conclude by hypothesis. ∎

From now on we may assume that each of the letters $1,\dots,d$ appears exactly twice in $\nu$. In particular, $d=2m$. As in the proof of Corollary A.9, we can assume, by induction on $\ell(\nu)$, that $P^{2}_{|M}=\det_{2}\nu_{|M}$ for every monomial $M\neq M_{\nu}$. In order to prove that the two polynomials coincide, it is enough to find a point $q\in\mathbb{R}^{2m}$ such that $M_{\nu}(q)\neq 0$ and $P^{2}(q)=\det_{2}\nu(q)$. Since we suppose that every entry of $\nu$ appears exactly twice, we can define $q\in\mathbb{R}^{2m}$ by

[TABLE]

The following is a straightforward consequence of Lemma 3.3.

Lemma A.10**.**

The matrix $\sigma^{(2)}(q)$ is skew-symmetric. If $i\neq j$, then

[TABLE]

To finish the proof it is enough to show that $P(q)^{2}=2^{d}\det_{2}(\nu)(q)$. As the second signature matrix is skew-symmetric, it is enough to prove that $P(q)=2^{d/2}\operatorname{Pf}_{2}(\nu)(q)$, where $\operatorname{Pf}$ is the Pfaffian. We note that when $\nu=(1,1,2,2,\dots,d,d)$ the claim is easy as both sides equal zero. The following two lemmas allow to reduce any $\nu$ to this case, finishing the proof.

Let $\nu_{i},\nu_{i+1}$ be two consecutive entries in $\nu$. Let $\nu^{\prime}$ be the sequence obtained from $\nu$ by switching $\nu_{i}$ and $\nu_{i+1}$, and let $q^{\prime}\in\mathbb{R}^{2m}$ be the corresponding point, defined in the same way we defined $q$. Let $\nu^{\prime\prime}$ be the sequence obtained from $\nu$ by removing both occurrences of the symbol $\nu_{i}$ and both occurrences of the symbol $\nu_{i+1}$. In a similar fashion we define $q^{\prime\prime}\in\mathbb{R}^{2m-4}$. By Remark A.1, we may assume without loss of generality that $\nu_{i}=d-1$ and $\nu_{i+1}=d$.

Lemma A.11**.**

Let

[TABLE]

In a similar way we define $e(d-1)$. Then

$\operatorname{Pf}_{2}(\nu)(q)=\operatorname{Pf}_{2}(\nu^{\prime})(q)+e(d-1)e(d)\operatorname{Pf}_{2}(\nu^{\prime\prime})(q^{\prime\prime})$* and* 2. 2.

$P_{\nu}(q)=P_{\nu^{\prime}}(q)+2e(d-1)e(d)P_{\nu^{\prime\prime}}(q^{\prime})$.

Proof.

The change from $\nu$ to $\nu^{\prime}$ changes the second signature matrix by replacing the lower right $2\times 2$ submatrix either from $\begin{pmatrix}0&0\\ 0&0\end{pmatrix}$ to $\pm\begin{pmatrix}0&1\\ -1&0\end{pmatrix}$ or the other way round. The formula follows from the standard Laplace expansion for Pfaffians. 2. 2.

The good underlinings that involve at most one of the exchanged $d-1$ and $d$ provide the same contribution both to $\nu$ and $\nu^{\prime}$. It remains to investigate the contribution of good underlinings containing both $d-1$ and $d$. These contribute to $\nu$ and $\nu^{\prime}$ with opposite signs and are exactly the contributions of $P_{\nu^{\prime\prime}}(q^{\prime})$. The sign depends only on the property if the underlined variables we forget are taken with plus or minus.∎

We conclude by induction on the number of permutations needed to transform $\nu$ to $1122\dots dd$ and the length of $\nu$.