Generalized Nowicki conjecture

TL;DR

This paper extends the proof of the Nowicki conjecture, describing the algebra of constants for certain derivations, providing explicit generators, relations, and a Gr"obner basis, thus deepening understanding of invariant rings in polynomial algebras.

Contribution

It generalizes previous results by explicitly presenting the algebra of invariants for derivations with polynomial images, including generators, relations, and Gr"obner basis structure.

Findings

Explicit presentation of the algebra of invariants as a quotient of a polynomial ring.

Identification of a reduced Gr"obner basis for the defining ideal.

Extension of the Nowicki conjecture to more general polynomial derivations.

Abstract

Let $B$ be an integral domain over a field $K$ of characteristic 0. The derivation $\delta$ of $B[Y_d]=B[y_1,\ldots,y_d]$ is elementary if $\delta(B)=0$ and $\delta(y_i)\in B$, $i=1,\ldots,d$. Then the elements $u_{ij}=\delta(y_i)y_j-\delta(y_j)y_i$, $1\leq i<j\leq d$, belong to the algebra $B[Y_d]^{\delta}$ of constants of $\delta$ and it is a natural question whether the $B$-algebra $B[Y_d]^{\delta}$ is generated by all $u_{ij}$. In this paper we consider the special case of $B=K[X_d]=K[x_1,\ldots,x_d]$. If $\delta(y_i)=x_i$, $i=1,\ldots,d$, this is the Nowicki conjecture from 1994 which was confirmed in several papers based on different methods. The case $\delta(y_i)=x_i^{n_i}$, $n_i>0$, $i=1,\ldots,d$, was handled by Khoury in the first proof of the Nowicki conjecture given by him in 2004. As a consequence of the proof of Kuroda in 2009 if $\delta(y_i)=f_i(x_i)$, for any nonconstant polynomials $f_i(x_i)$, $i=1,\ldots,d$, then $B[Y_d]^{\delta}=K[X_d,Y_d]^{\delta}$ is generated by $X_d$ and $U_d=\{u_{ij}=f_i(x_i)y_j-y_if_j(x_j)\mid 1\leq i<j\leq d\}$. In the present paper we have found a presentation of the algebra \[ K[X_d,Y_d]^{\delta}=K[X_d,U_d\mid R=S=0], \] \[ R=\{r(i,j,k,l)\mid 1\leq i<j<k<l\leq d\}, S=\{s(i,j,k)\mid 1\leq i<j<k\leq d\}, \] and a basis of $K[X_d,Y_d]^{\delta}$ as a vector space. As a corollary we have shown that the defining relations $R\cup S$ form the reduced Gr\"obner basis of the ideal which they generate with respect to a specific admissible order. This is an analogue of the result of Makar-Limanov and the author in their proof of the Nowicki conjecture in 2009.

Equations

$$K[X_d,Y_d{]}^{\delta }=K[X_d,U_d\mid R=S=0],$$

$$R=\{r(i,j,k,l)\mid 1\le i<j<k<l\le d\},S=\{s(i,j,k)\mid 1\le i<j<k\le d\},$$

$$\delta (a_1{a}_2)=\delta (a_1)a_2+a_1\delta (a_2),a_1,a_2\in A.$$

$$u_{ij}=\left|\begin{array}{cc}\delta (y_i)& \delta (y_j)\\ y_i& y_j\end{array}\right|=\delta (y_i)y_j-\delta (y_j)y_i,1\le i<j\le d,$$

$$\delta (y_i)=x_i,i=1,\dots ,d,$$

$$u_{ij}=x_i{y}_j-x_j{y}_i,1\le i<j\le d.$$

$$\delta (y_i)=x_i^{n_i},i=1,\dots ,d,$$

$$u_{ij}=x_i^{n_i}{y}_j-x_j^{n_j}{y}_i,1\le i<j\le d.$$

$$\Delta (y_i)=f_i(x_i),i=1,\dots ,d,$$

$$x_1^{r_1}\cdots x_d^{r_d},0\le r_i<m_i,i=1,\dots ,d.$$

$$x_1^{r_1}{f}_1^{n_1}(x_1)\cdots x_d^{r_d}{f}_d^{n_d}(x_d)$$

$$\Delta (y_i)=f_i(x_i),\Delta (x_i)=0,i=1,\dots ,d.$$

$$u_{ij}=f_i(x_i)y_j-f_j(x_j)y_i,1\le i<j\le d.$$

$$z_1=y_1,z_i=\alpha y_i-f_i(x_i)y_1,i=2,\dots ,d.$$

$$\Delta (z_1)=\alpha ,\Delta (z_i)=0,i=2,\dots ,d,\Delta (X_d)=0.$$

$$U_d=\{u_{ij}\mid 1\le i<j\le d\},$$

$$R=\{r(i,j,k,l)=u_{ij}{u}_{kl}-u_{ik}{u}_{jl}+u_{il}{u}_{jk}\mid 1\le i<j<k<l\le d\},$$

$$S=\{s(i,j,k)=f_i(x_i)u_{jk}-f_j(x_j)u_{ik}+f_k(x_k)u_{ij}\mid 1\le i<j<k\le d\}.$$

$$r(i,j,k,l)=\left|\begin{array}[]{cccc}f_{i}(x_{i})&f_{j}(x_{j})&f_{k}(x_{k})&f_{l}(x_{l})\\ y_{i}&y_{j}&y_{k}&y_{l}\\ f_{i}(x_{i})&f_{j}(x_{j})&f_{k}(x_{k})&f_{l}(x_{l})\\ y_{i}&y_{j}&y_{k}&y_{l}\\ \end{array}\right|$$

$$s(i,j,k)=\left|\begin{array}[]{ccc}f_{i}(x_{i})&f_{j}(x_{j})&f_{k}(x_{k})\\ f_{i}(x_{i})&f_{j}(x_{j})&f_{k}(x_{k})\\ y_{i}&y_{j}&y_{k}\\ \end{array}\right|$$

$$\pi :K[X_d,U_d]\to K[X_d,Y_d{]}^{\Delta }.$$

$$v=x_{i_1}\cdots x_{i_p}{u}_{j_1{k}_1}\cdots u_{j_q{k}_q}\text{ and }{v}^{\prime }=x_{i_1^{\prime }}\cdots x_{i_{p^{\prime }}^{\prime }}{u}_{j_1^{\prime }{k}_1^{\prime }}\cdots u_{j_{q^{\prime }}^{\prime }{k}_{q^{\prime }}^{\prime }},$$

$$\sum _{b=1}^q(k_b-j_b)>\sum _{b=1}^q(k_b^{\prime }-j_b^{\prime })$$

$$\sum _{b=1}^q(k_b-j_b)=\sum _{b=1}^q(k_b^{\prime }-j_b^{\prime })$$

$$\omega =(i_1,\dots ,i_p,j_1,\dots ,j_q,k_1,\dots ,q_d),{\omega }^{\prime }=(i_1^{\prime },\dots ,i_p^{\prime },j_1^{\prime },\dots ,j_q^{\prime },k_1^{\prime },\dots ,k_q^{\prime }),$$

$$v=x_1^{a_1}\cdots x_d^{a_d}{u}_{j_1{k}_1}\cdots u_{j_q{k}_q},1\le j_b<k_b\le d,$$

$$u_{ik}{u}_{jl},1\le i<j<k<l\le d,\text{ and }{x}_j^{m_j}{u}_{ik},1\le i<j<k\le d.$$

$$\pi (v)=x_1^{a_1}\cdots x_d^{a_d}\prod _{b=1}^q(f_{j_b}(x_{j_b})y_{k_b}-f_{k_b}(x_{k_b})y_{j_b}).$$

$$x_1^{a_1}{y}_1^{b_1}\cdots x_d^{a_d}{y}_d^{b_d}>x_1^{a_1^{\prime }}{y}_1^{b_1^{\prime }}\cdots x_d^{a_d^{\prime }}{y}_d^{b_d^{\prime }},$$

$$\text{lead}(\pi (v))=x_1^{a_1}\cdots x_d^{a_d}{x}_{j_1}^{m_{j_b}}{y}_{k_1}\cdots x_{j_1}^{m_{j_q}}{y}_{k_q}.$$

Full text

[TABLE]

[TABLE]

and a basis of $K[X_{d},Y_{d}]^{\delta}$ as a vector space. As a corollary we have shown that the defining relations $R\cup S$ form the reduced Gröbner basis of the ideal which they generate with respect to a specific admissible order. This is an analogue of the result of Makar-Limanov and the author in their proof of the Nowicki conjecture in 2009.

1. Introduction

In the present paper we consider only commutative algebras over a field $K$ of characteristic 0. A linear operator $\delta$ of an algebra $A$ is a derivation if it satisfies the Leibniz rule

[TABLE]

The kernel $A^{\delta}$ of $\delta$ is the algebra of constants of $\delta$. Let $B$ be an integral domain over $K$ and let $A=B[Y_{d}]=B[y_{1},\ldots,y_{d}]$. The derivation $\delta$ of $A$ is elementary if $\delta(B)=0$ and $\delta(y_{i})\in B$, $i=1,\ldots,d$. Then the determinants

[TABLE]

belong to $B[Y_{d}]^{\delta}$ and it is a natural question whether the $B$-algebra $B[Y_{d}]^{\delta}$ is generated by the elements (1). In the sequel we assume that $B=K[X_{d}]=K[x_{1},\ldots,x_{d}]$ (and $\delta(B)=0$). In the special case

[TABLE]

the finite generation of $A^{\delta}=B[Y_{d}]^{\delta}=K[X_{d},Y_{d}]^{\delta}$ follows from a more general result of Weitzenböck [12] in 1932. In 1994 Nowicki [10] conjectured that for $\delta$ from (2) the algebra $K[X_{d},Y_{d}]^{\delta}$ is generated by $X_{d}$ and

[TABLE]

This was confirmed in several papers based on different methods, see, e.g., [9] and [2] for details.

In the first proof of the Nowicki conjecture given in his Ph.D. thesis Khoury [6, 8] made one more step and established a result which gives an answer to a generalization of the Nowicki conjceture. If $n_{i}$, $i=1,\ldots,d$, are positive integers, and the derivation $\delta$ of $B[Y_{d}]=K[X_{d},Y_{d}]$ is defined by

[TABLE]

then the algebra $K[X_{d},Y_{d}]^{\delta}$ of constants of $\delta$ is generated again by $X_{d}$ and

[TABLE]

The most general result in this direction belongs to Kuroda [9].

Theorem 1.1**.**

Let $B$ be an integral domain over $K$ and let $\delta$ be an elementary derivation of $B[Y_{d}]$ such that $f_{i}=\delta(y_{i})$, $i=1,\ldots,d$, are algebraically independent over $K$. If $B$ is flat over $K[F_{d}]=K[f_{1},\ldots,f_{d}]$, then the $B$-algebra $B[Y_{d}]^{\delta}$ is generated by $u_{ij}=f_{i}y_{j}-f_{j}y_{i}$, $1\leq i<j\leq d$.

It seems that it is difficult to find further generalizations. Khoury [7] showed that the algebra of constants of the elementary derivation $\delta$ of $B[Y_{4}]$, $B=K[X_{3}]$, for $F_{4}=\{x_{1}^{2},x_{2}^{2},x_{3}^{2},x_{2}x_{3}\}$ is finitely generated but cannot be generated by expressions which are linear in $Y_{4}$. Also, many of the modern counterexamples to the Fourteenth Hilbert problem are constructed in terms of elementary derivations, see, e.g., the surveys by Freudenburg [5] and Nowicki [11].

It is easy to see that Theorem 1.1 holds for the elementary derivation $\Delta$ of $B[Y_{d}]$, $B=K[X_{d}]$, and

[TABLE]

where $f_{i}(x_{i})$ is a nonconstant polynomial in the variable $x_{i}$, $i=1,\ldots,d$. But there is an essential difference between the Nowicki conjecture and this result. The Nowicki conjecture is equivalent to a statement of classical invariant theory. When the polynomials $f_{i}(x_{i})$ are not linear, we cannot see how to restate the result in the language of invariant theory. In the present paper we apply the methods developed in our proof with Makar-Limanov [3] of the Nowicki conjecture and find a presentation of the algebra $K[X_{d},Y_{d}]^{\Delta}$ and a basis of $K[X_{d},Y_{d}]^{\Delta}$ as a vector space. As a consequence we show that our defining relations form the reduced Gröbner basis of the ideal which they generate with respect to a specific admissible order.

2. Generators of the algebra of constants

We shall need the following easy lemma. We include the proof for self-containedness of the exposition.

Lemma 2.1**.**

Let $f_{i}=f_{i}(x_{i})$, $i=1,\ldots,d$, be nonconstant polynomials. Then $K[X_{d}]$ is a free $K[f_{1},\ldots,f_{d}]$-module.

Proof.

Let $m_{i}=\deg(f_{i})$ be the degree of $f_{i}$. Every monomial $x_{i}^{m}$ can be written as a linear combination of polynomials $q_{n}(x_{i})f_{i}^{n}(x_{i})$, $\deg(q_{n})<m_{i}$, $n=0,1,2,\ldots$, and hence as a $K[f_{1},\ldots,f_{d}]$-module $K[X_{d}]$ is generated by the monomials

[TABLE]

The leading terms of the products

[TABLE]

with respect to the lexicographic order are equal to $x_{1}^{n_{1}m_{1}+r_{1}}\cdots x_{1}^{n_{d}m_{d}+r_{d}}$ and are pairwise different. Hence the polynomials (3) are linearly independent which implies that $K[X_{d}]$ is a free $K[f_{1},\ldots,f_{d}]$-module. ∎

Theorem 2.2**.**

Let $f_{i}=f_{i}(x_{i})$, $i=1,\ldots,d$, be nonconstant polynomials in one variable and let $\Delta$ be the derivation of $K[X_{d},Y_{d}]$ defined by

[TABLE]

Then the algebra $K[X_{d},Y_{d}]^{\Delta}$ of constants of $\Delta$ is generated by $X_{d}$ and the determinants from (1)

[TABLE]

Proof.

By [4, Corollary 6.6, p. 165] if a $B$-module $M$ is finitely generated then it is flat if and only if it is a summand of a free $B$-module. Hence by Lemma 2.1 $K[X_{d}]$ is a flat $K[f_{1},\ldots,f_{d}]$-module. Obviously $f_{i}(x_{i})$, $i=1,\ldots,d$, are algebraically independent over $K$ and Theorem 1.1 immediately gives that the algebra $K[X_{d},Y_{d}]^{\delta}$ is generated by $X_{d}$ and the polynomials (4). ∎

Remark 2.3**.**

If some of the polynomials $f_{i}(x_{i})$ in Theorem 2.2 is a constant, then the description of $K[X_{d},Y_{d}]^{\Delta}$ is trivial. Let, for example, $f_{1}(x_{1})=\alpha\in K\setminus\{0\}$. We replace the variables $Y_{d}$ by $Z_{d}$, where

[TABLE]

Then the definition of $\Delta$ becomes

[TABLE]

Since $\alpha\not=0$, we obtain that $K[X_{d},Z_{d}]^{\Delta}=K[X_{d},z_{2},\ldots,z_{d}]$.

3. The main result

In this section we follow our paper with Makar-Limanov [3] and use the methods developed there. Since we shall work with Gröbner bases, we refer, e.g., to the book by Adams and Loustaunau [1] for a background on the topic. We fix the degrees $m_{1},\ldots,m_{d}$ of the polynomials $f_{1}(x_{1}),\ldots,f_{d}(x_{d})$ and the set

[TABLE]

where the elements $u_{ij}$ are defined in (4).

Lemma 3.1**.**

*The subsets $X_{d}$ and $U_{d}$ of $K[X_{d},Y_{d}]$ satisfy the relations $R=S=0$, where *

[TABLE]

[TABLE]

Proof.

The annihilating of the relations (5) and (6) in $K[X_{d},Y_{d}]$ can be verified directly. Instead, the expansions of the determinants

[TABLE]

and

[TABLE]

relative to the first two rows and to the first row give, respectively, (5) and (6). ∎

Now we shall work in the polynomial algebra $K[X_{d},U_{d}]$. By Theorem 2.2 there is a canonical epimorphism

[TABLE]

Since there will be no misunderstanding, we shall use the same symbols $x_{i}\in X_{d}$ and $u_{jk}\in U_{d}$ for the generators of the polynomial algebra $K[X_{d},U_{d}]$ and their images under $\pi$ which generate the algebra of constants $K[X_{d},Y_{d}]^{\Delta}$.

Our first goal is to show that the kernel of $\pi$ is generated by the relations $r(i,j,k,l)$ and $s(i,j,k)$ from (5) and (6). We associate with every $u_{ij}\in U_{d}$ the open interval $(i,j)$. As in [3] we define an ordering of $K[X_{d},U_{d}]$ called degree–interval length–lexicographic order (DILL order). We order the monomials of $K[X_{d},U_{d}]$ first by the degree in $X_{d}$ and in $U_{d}$, then by the total length of the intervals associated with the participating variables $u_{ij}$ and finally lexicographically. If

[TABLE]

where $1\leq i_{1}\leq\cdots\leq i_{p}\leq d$, $1\leq j_{1}\leq\cdots\leq j_{q}\leq d$ and $1\leq k_{a}\leq k_{a+1}\leq d$ if $j_{a}=j_{a+1}$, with similar restrictions on $i^{\prime}_{a},j^{\prime}_{b},k^{\prime}_{b}$, we define $v\succ v^{\prime}$ if

(i) $p>p^{\prime}$ (we compare the degrees of $v$ and $v^{\prime}$ in $X_{d}$);

(ii) $p=p^{\prime}$ and $q>q^{\prime}$ (we compare the degrees of $v$ and $v^{\prime}$ in $U_{d}$);

(iii) $p=p^{\prime}$, $q=q^{\prime}$ and

[TABLE]

(we compare the total lengths of the intervals associated with $v$ and $v^{\prime}$);

(iv) $p=p^{\prime}$, $q=q^{\prime}$,

[TABLE]

and $\omega>\omega^{\prime}$ for the $(p+2q)$-tuples

[TABLE]

where $(a_{1},\ldots,a_{n})>(b_{1},\ldots,b_{n})$ if $a_{1}=b_{1},\ldots,a_{e}=b_{e}$, $a_{e+1}>b_{e+1}$ for some $e$ (we compare lexicographically $v$ and $v^{\prime}$).

Obviously the DILL-order is admissible, i.e., it is linear, satisfies the descending chain condition, and if $v\succ v^{\prime}$ for two monomials $v$ and $v^{\prime}$, then $vw\succ v^{\prime}w$ for all monomials $w$. For $0\not=f\in K[X_{d},U_{d}]$ we denote by $\overline{f}$ the leading monomial of $f$.

Lemma 3.2**.**

The set of normal words in $K[X_{d},U_{d}]$ with respect to DILL order and modulo the relations $R\cup S$ consists of the monomials

[TABLE]

such that

(i)* If $(j_{b},k_{b})\cap(j_{c},k_{c})\not=\emptyset$ for two different $u_{j_{b}k_{b}}$ and $u_{j_{c}k_{c}}$ in (8), then one of the intervals $(j_{b},k_{b})$ and $(j_{c},k_{c})$ is contained in the other;*

(ii)* If $i\in(j_{b},k_{b})$ for some $u_{j_{b}k_{b}}$ in (8), then $a_{i}<m_{i}$.*

As a vector space the algebra $K[X_{d},Y_{d}]^{\Delta}$ is spanned by the images under $\pi$ of the products (8).

Proof.

The leading monomials with respect to the DILL order of $r(i,j,k,l)$ and $s(i,j,k)$ from (5) and (6) are, respectively,

[TABLE]

(i) Let $u_{j_{b}k_{b}}u_{j_{c}k_{c}}$ divide the monomial $v$, and let $j_{b}\leq j_{c}$ and if $j_{b}=j_{c}$, then $k_{b}\leq k_{c}$. If the intervals $(j_{b},k_{b})$ and $(j_{c},k_{c})$ have a nontrivial intersection and are not contained in each other, then $j_{b}<j_{c}<k_{b}<k_{c}$. Hence the monomial $u_{j_{b}k_{b}}u_{j_{c}k_{c}}$ is the leading term of $r(j_{b},j_{c},k_{b},k_{c})$. In this way $v$ is not a normal word and this proves (i).

(ii) If the monomial $v$ is a normal word and $x_{i}^{a_{i}}u_{j_{b}k_{b}}$ divides $v$ with $i\in(j_{b},k_{b})$, then $x_{i}^{a_{i}}u_{j_{b}k_{b}}$ is not divisible by a monomial from (9) and hence $a_{i}<m_{i}$. This proves (ii).

Let us take some set of polynomials $W=\{w_{1},\ldots,w_{n}\}\subset K[X_{d},U_{d}]$ which are in the kernel of $\pi$ from (7). The images in $K[X_{d},Y_{d}]^{\Delta}$ of the normal words in $K[X_{d},U_{d}]$ with respect to $W$ span the vector space $K[X_{d},Y_{d}]^{\Delta}$. By Lemma 3.1 the set of relations $R\cup S$ from (5) and (6) belong to the kernel of $\pi$. Hence $K[X_{d},Y_{d}]^{\Delta}$ is spanned by the images of the normal words from (8). ∎

Lemma 3.3**.**

The images under the epimorphism $\pi$ from (7) of the normal words (8) from Lemma 3.2 form a basis of the vector space $K[X_{d},Y_{d}]^{\Delta}$.

Proof.

Let $v=x_{1}^{a_{1}}\cdots x_{d}^{a_{d}}u_{j_{1}k_{1}}\cdots u_{j_{q}k_{q}}\in K[X_{d},U_{d}]$ be a normal word. Then

[TABLE]

We shall follow the proof of [3, Step 3 in the proof of Theorem 5]. We compare the monomials in $K[X_{d},Y_{d}]=K[x_{1},y_{1},\ldots,x_{d},y_{d}]$ lexicographically:

[TABLE]

if $(a_{1},b_{1},\ldots,a_{d},b_{d})>(a_{1}^{\prime},b_{1}^{\prime},\ldots,a_{d}^{\prime},b_{d}^{\prime})$ if the usual lexicographic order. If $j_{b}<k_{b}$, then the leading monomials of $f_{j_{b}}(x_{j_{b}})y_{k_{b}}$ and $f_{k_{b}}(x_{k_{b}})y_{j_{b}}$ are $x_{j_{b}}^{m_{j_{b}}}y_{k_{b}}$ and $y_{j_{b}}x_{k_{b}}^{m_{k_{b}}}$. Hence $x_{j_{b}}^{m_{j_{b}}}y_{k_{b}}>y_{j_{b}}x_{k_{b}}^{m_{k_{b}}}$ and the leading monomial of (10) is

[TABLE]

We shall show that we can recover the normal word $v$ from the leading monomial $\text{lead}(\pi(v))$. Hence the monomials $\text{lead}(\pi(v))$ are pairwise different and the polynomials $\pi(v)$ are linearly independent. By Lemma 3.2 this implies that $\pi(v)$ form a basis of the vector space $K[X_{d},Y_{d}]^{\Delta}$.

It is sufficient to compare those $\pi(v)$ which are of the same degree in $Y_{d}$. As in [3] we use induction on the degree with respect to $Y_{d}$. Since $\deg_{U_{d}}(v)=\deg_{Y_{d}}(\text{lead}(\pi(v)))$, if $\deg_{Y_{d}}(\text{lead}(\pi(v)))=0$, then $\deg_{U_{d}}(v)=0$ and $v$ coincides with $\text{lead}(\pi(v))$. Now let $\deg_{Y_{d}}(\text{lead}(\pi(v)))>0$. We rewrite (11) in the form

[TABLE]

The variable $y_{k}$ participates in $\text{lead}(\pi(v))$ because $u_{jk}$ participates in $v$ from (8) for some $j<k$ and $p_{j}\geq m_{j}$. We choose the maximal $i$ such that $i<k$ and $p_{i}\geq m_{i}$. Lemma 3.2 implies that $i$ does not belong to the ideal $(j,k)$. Hence $i=j$ and the factor $u_{jk}=u_{ik}$ of $v$ is uniquely determined by the form (12) of $\text{lead}(\pi(v))$. In this way $v=u_{ik}v_{1}$ for some normal word $v_{1}\in K[X_{d},U_{d}]$ and

[TABLE]

By inductive arguments $\text{lead}(\pi(v_{1}))$ determines $v_{1}$ and this completes the proof. ∎

The following theorem is the main result of the paper.

Theorem 3.4**.**

Let $\Delta$ be the derivation of $K[X_{d},Y_{d}]$ defined by

[TABLE]

where $f_{i}(x_{i})$ are polynomials of positive degree. Then

(i)* The algebra of constants $K[X_{d},Y_{d}]^{\Delta}$ has the presentation*

[TABLE]

where

[TABLE]

and the ideal $(R,S)$ is generated by $R$ and $S$ from (5) and (6).

(ii)* The set $R\cup S$ is a reduced Gröbner basis of the ideal $(R,S)$ with respect to the DILL order of $K[X_{d},U_{d}]$.*

(iii)* The basis of the vector space $K[X_{d},Y_{d}]^{\Delta}$ consists of the images in $K[X_{d},Y_{d}]^{\Delta}$ of the normal words $v$ from Lemma 3.2.*

Proof.

By Lemma 3.1 $R\cup S$ belongs to the kernel of the epimorphism $\pi$ from (7). A subset of the ideal $\ker(\pi)$ is its Gröbner basis if and only if the set of normal words forms a basis of the factor algebra $K[X_{d},U_{d}]/\ker(\pi)\cong K[X_{d},Y_{d}]^{\Delta}$. Hence the statements (i), (ii), and (iii) follow immediately from Lemma 3.3. Since the leading terms of the relations $r(i,j,k,l)\in R$ and $s(i,j,k)\in S$ with respect to the DILL order do not divide the monomials participating in the other relations in $R$ and $S$, we conclude that the Gröbner basis $R\cup S$ is reduced. ∎