Some results on Gröbner bases over commutative rings

Abstract

Let F be a finite set of polynomials in A[x], where A is a commutative ring and x is a single variable. This paper is concerned with what properties are imposed on the coefficients of these polynomials if F is assumed to be a Gröbner basis. When A = R[y] for some commutative ring R, we characterize Gröbner bases in R[y, x] in terms of Gröbner bases in R[y][x] and Grobner bases in R[y]. We then address the question of lifting Gröbner bases from A to A[x] by examining the relationship between Szekeres bases and Gröbner bases. Finally we show that if A is a UFD, then, if the elements of F form a Grobner basis and are relatively prime, the same is true of the leading coefficients of the polynomials in F.