Factorization in \(K[S]\) (Q2785913)

The paper under review is devoted to give an upper bound of the elasticity of the semigroup ring \(K[S]\) of a numerical semigroup \(S\), \(K\) being a field. The elasticity of an atomic domain \(D\) is defined to be the number \(\rho(D):=\sup\{m/n\mid x_1\dots x_m=y_1\dots y_n\}\) where \(x_i\), \(y_j\) are irreducible elements in \(D\). The aim of the paper is therefore to study lengths of factorization in \(K[S]\).NEWLINENEWLINENEWLINEThe bound is given in terms of the Frobenius number of \(S\), \(g(S):=\max\{n\in\mathbb{Z}_+\mid n\notin S\}\) and the Davenport constant of certain finite abelian group associated with \(K\) and \(S\). The Davenport constant of a finite abelian group is the least positive integer \(d\) such that for each sequence \(S\subset G\) with \(\text{card}(S)=d\), some nonempty sequence of \(S\) has sum 0.NEWLINENEWLINENEWLINEFurthermore, the elasticity of \(K[S]\) is explicitly computed (and equal to the above mentioned upper bound) in the case when \(X^{g(S)+n_1}\in K[S]\) is irreducible in \(K[S]\), \(n_1\) being the first generator of \(S\), or equivalently in the case when \(g(S)+n_1=n_r\), where \(n_r\) is the last generator of \(S\).NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].