Abstract
Let a,b and n>1 be three positive integers such that a and ∑j=0n−1bj are relatively prime. In this paper, we prove that the toric ideal I associated to the submonoid of N generated by {∑j=0n−1bj}∪{∑j=0n−1bj+a∑j=0i−2bj∣i=2,…,n} is determinantal. Moreover, we prove that for n>3, the ideal I has a unique minimal system of generators if and only if a<b−1.
Highlights
Let k be a field and let A = { a1, . . . , an } be a set of positive integers
We prove that the toric ideal I associated to the submonoid of N generated by {∑nj=−01 b j } ∪
(Theorem 1) and, applying ([4], Corollary 14), we conclude that for n > 3, the ideal I has a unique minimal system of generators if and only if and a < b − 1 (Corollary 2)
Summary
This intermediate result (Proposition 1) has its own interest, as it exhibits another family of semigroup ideals that are determinantal and have unique minimal system of binomial generators (Corollary 1). The explicit description of minimal systems of binomial generators of monomial curves, and in a broader context of toric ideals, is a long-established research topic since. Despite of not being the aim this paper, the study of the defining ideal of monomials curves have its own interest for applications to other areas such as linear programming (see, e.g., [11]), coding theory (see, e.g., [12] or algebraic statistics, where the minimal systems of bionomial generators are called Markov bases and the uniqueness property has special consideration (see [13])
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
