Generalized Nowicki conjecture

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.