On the numbers of generators of monomial ideals
The author had earlier proved [ibid. 55, No. 6, 533-536 (1990; Zbl 0736.13011)] a bound for the number of generators of a radical monomial ideal in a polynomial ring. In this paper he generalises it to the case of arbitrary monomial ideals. The basic idea is based on a method of \textit{P. Schwartu} [Thesis, Brandeis Univ. (Waltham/MA, USA) 1982], by which the problem can be easily reduced to the case of radical ideals. If \(f=\prod^ n_ 1x_ i^{a_ i}\) is a monomial in a polynomial ring in the variables \(x_ i\), then we take the polynomial ring in \(d=\Sigma a_ i\) variables \(x_ i^{(j_ i)}\) where \(i\) runs over 1 to \(n\), and \(j_ i\) runs from 1 to \(a_ i\). We get a homomorphism from this new polynomial ring to the first one in an obvious way (by sending \(x_ i^{(j_ i)}\) to \(x_ i)\) and a reduced polynomial goes to \(f\). Thus computations can be made in this new polynomial ring for a reduced monomial ideal.
number of generators of a radical monomial ideal in a polynomial ring
scientific article; zbMATH DE number 62261