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Ignacio García-Marco (Universidad de la Laguna)

An algorithm for checking whether a simplicial toric ideal is a complete intersection

Let kbe an arbitrary field and k[x] = k[x1, … ,xn] and k[t] = k[t1, …, tm] two polynomial rings over k. Let A = {a1, …, an} be a set of nonzero vectors in Nm; each vector ai = (ai1, ... , aim) corresponds to a monomial tai = tai1 …taim m in k[t]. The kernel of the homomorphism of k-algebras f: k [x] ->k[t]; xi -> tai is called a toric ideal and will be denoted by IA. It is an A-homogeneous binomial ideal, i.e., if one sets the A-degree of a monomial  xa Îk[x] as a1a1 + …+anan ÎNm, and says that a polynomial fÎk [x] is A-homogeneous if its monomials have the same A-degree, then IA is generated by A-homogeneous binomials.

The ideal IA is a complete intersection if there exists a system of A-homogeneous binomials g1,…, gs such that IA= (g1,…, gs) , where s = n– rk (ZA).

If n> m and A = {d1e1,…, dmem; am+1,…, an}, where {e1, …, em} is the canonical basis of Zm, the toric ideal IA is said to be a simplicial toric ideal.

The purpose of this work is to provide and implement an algorithm for determining whether a simplicial toric ideal IA is a complete intersection without having to compute a minimal system of A-homogeneous generators of IA. This is a joint work with Isabel Bermejo.

© 3º Encontro Ibérico de Matemática :: 2010