Generalized Nowicki conjecture

Let $B$ be an integral domain over a field $K$ of characteristic 0. The derivation $\delta$ of $B[Y_d]=B[y_1,\dots ,y_d]$ is elementary if $\delta (B)=0$ and $\delta (y_i)\in B$, $i=1,\dots ,d$. Then the elements $u_{ij}=\delta (y_i)y_j-\delta (y_j)y_i$, $1\le i<j\le d$, 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 $B=K[X_d]=K[x_1,\dots ,x_d]$. If $\delta (y_i)=x_i$, $i=1,\dots ,d$, this is the Nowicki conjecture from 1994 which was confirmed in several papers based on different methods. The case $\delta (y_i)=x_i^{n_i}$, $n_i>0$, $i=1,\dots ,d$, 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 $f_i(x_i)$, $i=1,\dots ,d$, then $B[Y_d{]}^{\delta }=K[X_d,Y_d{]}^{\delta }$ is generated by $X_d$ and $U_d=\{u_{ij}=f_i(x_i)y_j-y_i{f}_j(x_j)\mid 1\le i<j\le d\}$. In the present paper we have found a presentation of the algebra $$K[X_d,Y_d{]}^{\delta }=K[X_d,U_d\mid R=S=0],$$ $$R=\{r(i,j,k,l)\mid 1\le i<j<k<l\le d\},S=\{s(i,j,k)\mid 1\le i<j<k\le d\},$$ 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.