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...

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...

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...

A straightforward way to represent multivariate polynomial in software is to implement them recursively as univariate polynomials over a polynomial ring. This is especially common in an object oriented context. We present a short algorithm which maps polynomials from one polynomial ring to another polynomial ring where the order of variables is permuted. This algorithm uses the recursive repres...

1. Amitsur in his paper on Finite Dimensional Central Division Algebras [l] has proved that in a division ring D with center C, (P: C) 5= ra2 < =o if and only if every primitive homomorphic image of a polynomial ring P[x] is a complete matrix ring Ah, h^n, over a division ring A. Equivalently speaking, a division ring is finite dimensional over its center if and only if the polynomial ring over...

Astract We show here that some simple combinatorial facts concerning arrangements of pebbles on an n×n board have surprising consequences in the study of expansions in the polynomial ring Q[x 1 , x 2 ,. .. , x n ]. In particular in this manner we obtain a purely combinatorial proof of an identity of Lascoux-Schutzenberger given in Funkt. Anal 21 (1987) 77-78.