On Fr\"oberg-Macaulay conjectures for algebras

TL;DR

This paper explores variants of Macaulay's theorem and Fr"oberg's conjecture for subalgebras of polynomial rings, presenting questions, partial results, and examples to understand their Hilbert functions.

Contribution

It introduces new questions and partial results on the Hilbert functions of subalgebras generated by forms, extending classical conjectures from ideals to subalgebras.

Findings

Partial results on Hilbert functions of subalgebras

Examples illustrating the behavior of these functions

Formulation of new questions related to the conjectures

Abstract

Macaulay's theorem and Fr\"oberg's conjecture deal with the Hilbert function of homogeneous ideals in polynomial rings $S$ over a field $K$. In this short note we present some questions related to variants of Macaulay's theorem and Fr\"oberg's conjecture for $K$-subalgebras of polynomial rings. In details, given a subspace $V$ of forms of degree $d$ we consider the $K$-subalgebra $K[V]$ of $S$ generated by $V$. What can be said about Hilbert function of $K[V]$? The analogy with the ideal case suggests several questions. To state them we start by recalling Macaulay's theorem, Fr\"oberg's conjecture and Gotzmann's persistence theorem for ideals. Then we presents the variants for $K$-subalgebras along with some partial results and examples.

Equations

$$\dim S_{j}L\le \dim S_{j}V$$

$$\dim S_{1}L=\dim S_1\mathrm{Lex}(u,S_d),$$

$$\dim S_{1}L\le \dim S_{1}V,$$

$$\dim S_{j}L\le \dim S_{j}V,$$

$$\dim S_{j}V\le \dim S_{j}W.$$

$$\sum c_i{z}^i={\left[\frac{(1-z^d{)}^u}{(1-z{)}^n}\right]}_{+}$$

$$L(n,d,u,j)=\min \{\dim V^j:V\in G(u,S_d)\}$$

$$M(n,d,u,j)=\max \{\dim V^j:V\in G(u,S_d)\}.$$

$$L(n,d,u,j)=\dim W^j$$

$$\begin{array}[]{rlcll}1)&\operatorname{St}\{xy^{3},x^{2}z^{2}\}&=&\langle xy^{3},x^{2}z^{2},x^{2}yz,x^{2}y^{2},x^{3}z,x^{3}y,x^{4}\rangle&\mbox{ -- the Lex Segment}\\ 2)&\operatorname{St}\{xy^{2}z\}&=&\langle xy^{2}z,xy^{3},x^{2}yz,x^{2}y^{2},x^{3}z,x^{3}y,x^{4}\rangle\\ 3)&\operatorname{St}\{y^{4},x^{2}yz\}&=&\langle y^{4},xy^{3},x^{2}yz,x^{2}y^{2},x^{3}z,x^{3}y,x^{4}\rangle&\mbox{ -- the RevLex Segment }\end{array}$$

$$\begin{array}[]{l}W_{1}=\operatorname{St}\{x^{2}z^{2},xy^{3}z\}=\langle x^{5},x^{4}y,x^{4}z,x^{3}y^{2},x^{3}yz,x^{3}z^{2},x^{2}y^{3},x^{2}y^{2}z,x^{2}yz^{2},x^{2}z^{3},xy^{4},xy^{3}z\rangle,\\ W_{2}=\operatorname{St}\{xy^{2}z^{2}\}=\langle x^{5},x^{4}y,x^{4}z,x^{3}y^{2},x^{3}yz,x^{3}z^{2},x^{2}y^{3},x^{2}y^{2}z,x^{2}yz^{2},xy^{4},xy^{3}z,xy^{2}z^{2}\rangle,\\ W_{3}=\operatorname{St}\{x^{2}z^{3},y^{5}\}=\langle x^{5},x^{4}y,x^{4}z,x^{3}y^{2},x^{3}yz,x^{3}z^{2},x^{2}y^{3},x^{2}y^{2}z,x^{2}yz^{2},x^{2}z^{3},xy^{4},y^{5}\rangle,\\ W_{4}=\operatorname{St}\{x^{2}z^{2}y,xy^{3}z,y^{5}\}=\langle x^{5},x^{4}y,x^{4}z,x^{3}y^{2},x^{3}yz,x^{3}z^{2},x^{2}y^{3},x^{2}y^{2}z,x^{2}yz^{2},xy^{4},xy^{3}z,y^{5}\rangle,\\ W_{5}=\operatorname{St}\{x^{3}z^{2},y^{4}z\}=\langle x^{5},x^{4}y,x^{4}z,x^{3}y^{2},x^{3}yz,x^{3}z^{2},x^{2}y^{3},x^{2}y^{2}z,xy^{4},xy^{3}z,y^{5},y^{4}z\rangle.\end{array}$$

$$\begin{array}[]{rl}\operatorname{HS}_{K[W_{1}]}(z)=\operatorname{HS}_{K[W_{5}]}(z)=&\displaystyle{\frac{1+9z+3z^{2}}{(1-z)^{3}}},\\ \operatorname{HS}_{K[W_{2}]}(z)=\operatorname{HS}_{K[W_{4}]}(z)=&\displaystyle{\frac{1+9z+2z^{2}}{(1-z)^{3}}},\\ \operatorname{HS}_{K[W_{3}]}(z)=&\displaystyle{\frac{1+9z+5z^{2}}{(1-z)^{3}}}.\\ \end{array}$$

$$M(n,d,u,j)\le \min \left\{\dim S_{jd},\left(\genfrac{}{}{0ex}{}{u-1+j}{u-1}\right)\right\}$$

$$M(n,d,u,j)=\min \left\{\dim S_{jd},\left(\genfrac{}{}{0ex}{}{u-1+j}{u-1}\right)\right\}$$

$$M(4,2,8,2)\le 34<\min \left\{\dim S_4,\left(\genfrac{}{}{0ex}{}{7+2}{7}\right)\right\}=35.$$

$$W_0=⟨x_i{x}_j:1\le i<j\le 4⟩$$

$$F=a_1{x}_1^2+a_2{x}_2^2+a_3{x}_3^2+a_4{x}_4^2\text{mbox}andG=b_1{x}_1^2+b_2{x}_2^2+b_3{x}_3^2+b_4{x}_4^2$$

$$W=W_0+W_1.$$

$$\left(\begin{array}{cccc}{a}_1& a_2& a_3& a_4\\ b_1& b_2& b_3& b_4\end{array}\right)$$

$$\left(\begin{array}{cccc}{a}_1^2& a_2^2& a_3^2& a_4^2\\ b_1^2& b_2^2& b_3^2& b_4^2\\ a_1{b}_1& a_2{b}_2& a_3{b}_3& a_4{b}_4\end{array}\right)$$

Full text

The first author was partially supported by VR2013-4545 and the second author was partially supported by INdAM-GNSAGA

Introduction

Macaulay’s theorem and Fröberg’s conjecture deal with the Hilbert function of homogeneous ideals in polynomial rings $S$ over a field $K$. In this short note we present some questions related to variants of Macaulay’s theorem and Fröberg’s conjecture for $K$-subalgebras of polynomial rings. In details, given a subspace $V$ of forms of degree $d$ we consider the $K$-subalgebra $K[V]$ of $S$ generated by $V$. What can be said about Hilbert function of $K[V]$? The analogy with the ideal case suggests several questions. To state them we start by recalling Macaulay’s theorem, Fröberg’s conjecture and Gotzmann’s persistence theorem for ideals. Then we presents the variants for $K$-subalgebras along with some partial results and examples.

1. Macaulay’s theorem and Fröberg’s conjecture for ideals

Let $K$ be a field and $S=K[x_{1},\dots,x_{n}]$ be the polynomial ring equipped with its standard grading, i.e., with $\deg x_{i}=1$ for $i=1,\dots,n$. Then $S=\bigoplus_{j=0}^{\infty}S_{j}$ where $S_{j}$ denotes the $K$-vector space of homogeneous polynomials of degree $j$. Given positive integers $d,u$ such that $u\leq\dim S_{d}$ let $G(u,S_{d})$ be the Grassmannian of all $u$-dimensional $K$-subspaces of $S_{d}$. For a given subspace $V\in G(u,S_{d})$, the homogeneous components of the ideal $I=(V)$ of $S$ generated by $V$ are the vector spaces $S_{j}V$, i.e., the vector spaces generated by the elements $fg$ with $f\in S_{j}$ and $g\in V$.

Question**.**

What can be said about the dimension of $S_{j}V$ in terms of $u=\dim V$?

1.1. Lower bound:

Macaulay’s theorem on Hilbert functions [Macaulay] provides a lower bound for $\dim S_{j}V$ given $\dim V$. It asserts that there exists a subspace $L\in G(u,S_{d})$ such that

[TABLE]

for every $j$ and every $V\in G(u,S_{d})$. Furthermore $\dim S_{j}L$ can be expressed combinatorially in terms of $d$ and $u$ by means of the so-called Macaulay expansion, see [BrunsHerzog, Sury] for details. The vector space $L$ can be taken to be generated by the largest $u$ monomials of degree $d$ with respect to the lexicographic order. Such an $L$ is called the lex-segment (vector space) associated to the pair $d$ and $u$ and it is denoted by $\operatorname{Lex}(u,S_{d})$.

1.2. Persistence:

A vector space $L\in G(u,S_{d})$ is called Gotzmann if it satisfies

[TABLE]

i.e., if

[TABLE]

for all $V\in G(u,S_{d})$. Gotzmann’s persistence theorem [Gotzmann] asserts that if $L\in G(u,S_{d})$ is Gotzmann then $S_{1}L$ is Gotzmann as well. In particular if $L$ is Gotzmann one has

[TABLE]

for all $j$ and all $V\in G(u,S_{d})$.

1.3. Upper bound:

Clearly, the upper bound for $\dim S_{j}V$ is given by the $\dim S_{j}W$ for a “general” $W$ in $G(u,S_{d})$. More precisely, there exists a non-empty Zariski open subset $U$ of $G(u,S_{d})$ such that for every $V\in G(u,S_{d})$, for every $j\in{\mathbb{N}}$ and every $W\in U$ one has

[TABLE]

Fröberg’s conjecture predicts the values of the upper bound $\dim S_{j}W$. For a formal power series $f(z)=\sum_{i=0}^{\infty}f_{i}z^{i}\in{\mathbb{Z}}\llbracket z\rrbracket$ one denotes $[f(z)]_{+}$ the series $\sum_{i=0}^{\infty}g_{i}z^{i}$, where $g_{i}=f_{i}$ if $f_{j}\geq 0$ for all $j\leq i$ and $g_{i}=0$ otherwise. Given $n,u$ and $d$ one considers the formal power series:

[TABLE]

and then Fröberg’s conjecture asserts that $\dim S_{j}W=\dim S_{j+d}-c_{d+j}$ for all $j$. It is known to be true in these cases:

• (1)

$n\leq 3$ and any $u,d,j$, [Froberg, Anick],

• (2)

$u\leq n+1$ and any $d,j$, [Stanley],

• (3)

$j=1$ and any $n,u,d$, [HochsterLaksov]

and it remains open in general. See [Nenashev] for some recent contributions.

2. Macaulay’s theorem and Fröberg’s conjecture for

subalgebras

For any subspace $V\in G(u,S_{d})$ we can consider the $K$-subalgebra $K[V]\subseteq S$ generated by $V$. Indeed, $K[V]$ is the coordinate ring of the closure of the image of the rational map ${\mathbb{P}}^{n-1}\dashrightarrow{\mathbb{P}}^{u-1}$ associated to $V$.

The homogeneous component of degree $j$ of $K[V]$ is the vector space $V^{j}$, i.e., the $K$-subspace of $S_{jd}$ generated by the elements of the form $f_{1}\cdots f_{j}$ with $f_{1},\dots,f_{j}\in V$.

Question**.**

What can be said about the dimension of $V^{j}$? In other words, what can be said about the Hilbert function of the $K$-algebra $K[V]$?

Definition 2.1**.**

For positive integers, $n$, $d$, $u$ and $j$, define

[TABLE]

and

[TABLE]

2.1. Lower bound:

Recall that a monomial vector space $W$ is said to be strongly stable if $mx_{i}/x_{j}\in W$ for every monomial $m\in W$ and $i<j$ such that $x_{j}|m$. Intersections, sums and products of strongly stable vector spaces are strongly stable. Given monomials $m_{1},\dots,m_{c}\in S_{d}$ the smallest strongly stable vector space containing them is denoted by $\operatorname{St}(m_{1},\dots,m_{c})$ and it is called the strongly stable vector space generated by $m_{1},\dots,m_{c}$.

Proposition 2.2**.**

Given $n,d,u$ and $j$ there exists a strongly stable vector space $W\in G(u,S_{d})$ such that

[TABLE]

independently of the field $K$.

Proof.

Given a term order $<$ on $S$ for every $V\in G(u,S_{d})$ one has $\operatorname{in}(V)^{j}\subseteq\operatorname{in}(V^{j})$ for every $j$. Hence one has $\dim V_{0}^{j}\leq\dim V^{j}$ where $V_{0}=\operatorname{in}(V)$. Therefore the lower bound $L(n,d,u,j)$ is achieved by a monomial vector space. Comparing the vector space dimension of monomial algebras is a combinatorial problem and hence we may assume the base field has characteristic [math]. Applying a general change of coordinates, we may put $V$ in “generic coordinates” and hence $\operatorname{in}(V)$ is the generic initial vector space of $V$ with respect to some term order. Being such it is Borel fixed. Since the base field has characteristic [math], we have that $\operatorname{in}(V)$ is strongly stable. Therefore the lower bound $L(n,d,u,j)$ is achieved by a strongly stable vector space. ∎

Example 2.3**.**

For $n=3,d=4,u=7$ there are $3$ strongly stable vector spaces:

[TABLE]

In this case $2)$ and $3)$ turns out to give rational normal scrolls of type $(3,2)$ and $(4,1)$ respectively and they give the minimal possible Hilbert function in all values.

Example 2.4**.**

For $n=3$, $d=5$ and $u=12$, there are five strongly stable subspaces of $S_{d}$:

[TABLE]

In this case, neither the Lex segment, $W_{1}$, nor the RevLex segment, $W_{5}$, achieve the minimum Hilbert function. The Hilbert series are given by

[TABLE]

and the minimum turns out to be $L(3,5,12,j)=\dim W_{2}^{j}=\dim W_{4}^{j}=6j^{2}+5j+1$, for $j\geq 1$.

Questions 2.5**.**

• (1)

Does there exist a (strongly stable) subspace $W\in G(u,S_{d})$ such that $L(n,d,u,j)=\dim W^{j}$ for every $j$?

• (2)

Given $n,d,u,j$ can one characterize combinatorially the strongly stable subspace(s) $W$ with the property $L(n,d,u,j)=\dim W^{j}$?

• (3)

Persistence: Assume $W\in G(u,S_{d})$ satisfies $L(n,d,u,2)=\dim W^{2}$. Does it satisfies also $L(n,d,u,j)=\dim W^{j}$ for all $j$?

Remark 2.6**.**

For $n=2$ there exists only one strongly stable vector space in $G(u,S_{d})$, i.e. $\langle x^{d},x^{d-1}y,\dots,x^{d-u+1}y^{u-1}\rangle$ (which is both the Lex and RevLev segment) and the questions in 2.5 have all straightforward answers.

Remark 2.7**.**

It is proved in [DN] that Lex-segments, RevLex-segments and principal strongly stable vector spaces define normal and Koszul toric rings (in particular Cohen-Macaulay). Furthermore in [DFMSS] it is proved that a strongly stable vector spaces with two strongly stable generators define a Koszul toric ring. On the other hand, there are examples of strongly stable vector spaces with a non-Cohen-Macaulay and non-quadratic toric ring, see [BC, Example 1.3].

2.2. Upper bound:

As in the ideal case, the upper bound is achieved by a general subspace $W$, i.e., for $W$ in a non-empty Zariski open subset of $G(u,S_{d})$.

Question**.**

What can be said about the value $M(n,d,u,j)$?

Obviously,

[TABLE]

and the naive expectation is that equality holds in (1), i.e., if $f_{1},\dots,f_{u}$ are general forms of degree $d$, then the monomials of degree $j$ in the $f_{i}$’s are either linearly independent or they span $S_{jd}$. It turns out that in nature things are more complex than expected at first. First of all, if $u>n$ then equality in (1) would imply that for a generic $W$ one would have $W^{j}=S_{jd}$ for large $j$. This fact can be stated in terms of projections of the $d$-th Veronese variety: the projection associated to $W$ is an isomorphism. Recall that a generic linear projection of a smooth projective variety of dimension $m$ from some projective space where its embedded, into a projective space of dimension $c$ is an isomorphism if $c\geq 2m+1$. Hence we have that if $u\geq 2n$ then equality in (1) holds at least for large $j$. On the other hand, for $n+1\leq u<2n$ equality in (1) should not be expected unless one knows that the corresponding projection of the Veronese variety behaves in an unexpected way.

Summing up, the most natural question turns out to be:

Question**.**

Assume that $u\geq 2n$. Is it true that

[TABLE]

holds for all $j$?

The answer turns out to be negative as the following example shows:

Proposition 2.8**.**

For any space $W$ generated by eight quadrics in four variables the dimension of $W^{2}$ is at most $34$ independently of the base field $K$. That is:

[TABLE]

Remark 2.9**.**

This assertion was proven in [COR]*Theorem. 2.4 using a computer algebra calculation. Here we present a more conceptual argument.

Proof.

Firstly we may assume that $K$ has characteristic [math] and is algebraically closed. Secondly we may assume that $W$ is generic. The $8$-dimension space of quadrics $W$ is apolar to a $2$-dimension space of quadrics, call it $V$. A pair of generic quadrics can be put simultaneously in diagonal form, i.e., that $V$ is generated by $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}$ and $a_{1}x_{1}^{2}+a_{2}x_{2}^{2}+a_{3}x_{3}^{2}+a_{4}x_{4}^{2}$. See for example [Wonenburger]. Indeed, it is sufficient that $V$ contains a quadric of rank $4$ since that can be put into the form $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}$ and the other form can then be diagonalized preserving the first. As a consequence, after a change of coordinates $W$ contains $x_{i}x_{j}$ with $1\leq i<j\leq 4$. Since $(x_{1}x_{4})(x_{2}x_{3})=(x_{1}x_{2})(x_{3}x_{4})$ and $(x_{1}x_{3})(x_{2}x_{4})=(x_{1}x_{2})(x_{3}x_{4})$ we have at least two independent relations among the $36$ generators of $W^{2}$. Therefore $\dim W^{2}\leq 34$. ∎

More precisely one has:

Proposition 2.10**.**

One has $M(4,2,8,2)=34$ independently of the base field $K$.

Proof.

We have already argued that $M(4,2,8,2)\leq 34$. Therefore it is enough to describe an $8$-dimension space of quadrics $W$ in $4$ variables such that $\dim W^{2}=34$. We set

[TABLE]

and

[TABLE]

Then we set $W_{1}=\langle F,G\rangle$ and then

[TABLE]

We consider two conditions on the coefficients $a_{1},a_{2},\dots,b_{4}$:

Conditions**.**

(1) All the $2$-minors of

[TABLE]

are non-zero.

(2) The matrix

[TABLE]

has rank $3$.

We observe that $W_{0}^{2}$ is generated by the $19$ monomials of degree $4$ and largest exponent $\leq 2$. Then we note that if $W_{1}$ contains a quadric $q$ supported on $x_{i}^{2},x_{j}^{2}$ and $x_{h}^{2}$ with $i,j,h$ distinct and $k\not\in\{i,j,h\}$ then $x_{k}x_{i}q=x_{k}x_{i}^{3}\mod(W_{0}^{2})$ and similarly for $j$ and $h$. This implies that if Condition (1) holds then $W_{0}^{2}+W_{0}W_{1}$ is generated by the $31$ monomials different from $x_{1}^{4},\dots,x_{4}^{4}$. Assuming that Condition (1) holds, we have that the matrix representing $F^{2},G^{2},FG$ in $S_{4}/W_{0}^{2}+W_{0}W_{1}$ is exactly the one appearing in Condition (2). Then are $F^{2},G^{2},FG$ are linearly independent mod $W_{0}^{2}+W_{0}W_{1}$ if and only if Condition (2) holds. Summing up, if Conditions (1) and (2) hold then $\dim W^{2}=34$. Finally we observe that for $F=x_{1}^{2}+x_{3}^{2}+x_{4}^{2}$ and $G=x_{2}^{2}+\alpha x_{3}^{2}+x_{4}^{2}$ the conditions (1) and (2) are satisfied provided $\alpha\neq 0$ and $\alpha\neq 1$. Hence this (conceptual) argument works for any field but ${\mathbb{Z}}/2{\mathbb{Z}}$. Over ${\mathbb{Z}}/2{\mathbb{Z}}$ one can consider the space $W$ generated by $x_{1}^{2},x_{2}^{2},x_{3}^{2},x_{1}x_{3},x_{2}x_{4},x_{3}x_{4},x_{2}x_{3}+x_{1}x_{4},x_{1}x_{2}+x_{4}^{2}$ and check with the help of a computer algebra system that $\dim W^{2}=34$. ∎

As far as we know the case discussed in Proposition 2.8 is the only known case where $u\geq 2n$ and the actual value of $M(n,d,u,j)$ is smaller than $\min\left\{\dim S_{jd},\binom{u-1+j}{u-1}\right\}$.

Acknowledgements: We thank Winfred Bruns and David Eisenbud for helpful discussions on the material of this paper.