Some results on Gröbner bases over commutative rings
Let \(R\) be a commutative ring with identity. Fix a term order on the set of monomials in \(R[x_ 1,\ldots,x_ m]\) (the ring of polynomials). A set \(F \subseteq R[x_ 1,\ldots,x_ m]\) is defined as usual a Gröbner basis (with coefficients in \(R\)) if \(\text{Lt} (F)=\text{Lt}(\langle F \rangle)\), i.e. if the ideal generated by the leading terms of the polynomials in \(F\) coincides with the set (ideal) of all the terms of the polynomials of the ideal generated by \(F\). Fix now two sets of variables: \(y:=\{y_ 1,\ldots,y_ n\}\) and \(x:=\{x_ 1,\ldots,x_ m\}\) and let \(R[x,y]\) denote the polynomial ring in the variables \(x\) and \(y\) respectively. Assume that the monomials in \(y\) (in \(x\)) have a term order denoted by \(<_ y\) (\(<_ x\) resp.); then one can define a term order on the monomials in \(y\) and \(x\) by: \[ y_ 1x_ 1 < y_ 2x_ 2 \quad\text{if}\quad x_ 1<_ x x_ 2 \;\text{ or } \;\;x_ 1 = x_ 2 \;\;\text{ and } \;y_ 1<_ y y_ 2, \] where \(y_ 1,y_ 2\) (\(x_ 1,x_ 2\)) are monomials in \(y\) (in \(x\) resp.). Let \(F:=\{f_ 1,\ldots,f_ t\}\subseteq R[x,y]\). Set \(G:=\{\text{lc}_ x(f_ 1), \ldots,\text{lc}_ x(f_ t) \}\), where \(\text{lc}_ x(f)\) is the leading coefficient of \(f\) as a polynomial with coefficients in \(R[y]\) with respect to the term order \(<_ x\). The paper analyzes the following problems: When is \(F\) a Gröbner basis with coefficients in \(R\)? When is \(F\) a Gröbner basis with coefficients in \(R[y]\)? When is \(G\) a Gröbner basis in \(R[y]\) with coefficients in \(R\)? Several results concerning these problems are established. In particular: Assume \(m=1\), i.e. \(x=\{x_ 1\}\). Then \(F\) is a Gröbner basis in \(R[x,y]\) with coefficients in \(R\) if and only if the following two conditions are satisfied: (i) \(F\) is a Gröbner basis in \(R[x,y]\) with coefficients in \(R[x]\) and (ii) for every \(i=0,\ldots,s\), \({\mathcal G}_ i\) is a Gröbner basis in \(R[y]\) with coefficients in \(R\) (where \({\mathcal G}_ i:= \{\text{lc}_ x (f) |\;f\in F_ j, \;j=1,\ldots, i \}\) and where \(F_ 0, \ldots, F_ s\) are the partition of \(F\) such that \(F_ 0\) contains all the polynomials of \(F\) whose leading term in \(<_ x\) is lowest, \(F_ 1\) are the polynomials of \(F\) whose leading term is lowest among those not in \(F_ 0\), and so on). In section 3 the notion of Szekeres basis is introduced for a set of polynomials in \(R[x]\) (in one variable \(x\) over a ring \(R\)) and it is shown that a set of polynomials is a Szekeres basis if and only if it is a Gröbner basis. In the last section it is shown that if \(R\) is a UFD and if \(F\subseteq R[x]\) is a Gröbner basis such that the greatest common divisor of the polynomials of \(F\) is \(1\), then the same is true for the leading coefficients of the polynomials in \(F\).
scientific article; zbMATH DE number 92592