We define reduced Gröbner bases in polynomial rings over a polynomial ring and introduce an algorithm for computing them. There exist some algorithms for computing Gröbner bases in polynomial rings over a polynomial ring. However, we cannot obtain the reduced Gröbner bases by these algorithms. In this paper we propose a new notion of reduced Gröbner bases in polynomial rings over a polynomial r...

In this paper, we study an irreducible decomposition structure of the $$\mathcal {D}$$ -module direct image $$\pi _+(\mathcal {O}_{\mathbb {c}^n})$$ for finite map : \mathbb {C}^n \rightarrow {C}^n/ ({\mathcal {S}_{n_1}\times \cdots \times \mathcal {S}_{n_r}}).$$ We explicitly construct simple components {C}^n})$$ by providing their generators and multiplicities. Using equivalence categories hi...

Abstract We propose a new method to tackle the integrability problem for evolutionary differential–difference equations of arbitrary order. It enables us produce necessary conditions, determine whether given equation is integrable or not, and advance in classification equations. define develop symbolic representation difference polynomial ring, operators formal series. In order formulate we int...

Let $R$ be a reversible ring which is $alpha$-compatible for an endomorphism $alpha$ of $R$ and $f(X)=a_0+a_1X+cdots+a_nX^n$ be a nonzero skew polynomial in $R[X;alpha]$. It is proved that if there exists a nonzero skew polynomial $g(X)=b_0+b_1X+cdots+b_mX^m$ in $R[X;alpha]$ such that $g(X)f(X)=c$ is a constant in $R$, then $b_0a_0=c$ and there exist nonzero elements $a$ and $r$ in $R$ such tha...

The first essential ingredient to build up Stein’s method for a continuous target distribution is identify so-called Stein operator, namely linear differential operator with polynomial coefficients. In this paper, we introduce the notion of algebraic operators (see Definition 3.4), and provide novel find all given order degree random variable form Y=h(X), where X=(X1,…,Xd) has i.i.d. standard G...

Consider two differential operators L1 = ∑ aid i and L2 = ∑ bjd j with coefficients in a differential field, say C(t) with d = ∂ ∂t for example. If the ai and bj are constants, the condition for the existence of a solution y of L1(y) = L2(y) = 0 is that the resultant in X of the polynomials (in C[X]) ∑ aiX i and ∑ bjX j is zero. A natural question is: how one could extend this for the case of n...

let $r$ be a reversible ring which is $alpha$-compatible for an endomorphism $alpha$ of $r$ and $f(x)=a_0+a_1x+cdots+a_nx^n$ be a nonzero skew polynomial in $r[x;alpha]$. it is proved that if there exists a nonzero skew polynomial $g(x)=b_0+b_1x+cdots+b_mx^m$ in $r[x;alpha]$ such that $g(x)f(x)=c$ is a constant in $r$, then $b_0a_0=c$ and there exist nonzero elements $a$ and $r$ in $r$ such tha...

We also give partial results on computation of cohomology groups on U for a locally constant sheaf of general rank and on computation of H(C \ Z,C) where Z is a general algebraic set. Our algorithm is based on computations of Gröbner bases in the ring of differential operators with polynomial coefficients, algorithms for functors in the theory of D-modules ([24] and [25]), and Grothendieck-Deli...