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[x_1, … ,x_n] and k[t] = k[t₁, …, t_m] two polynomial rings over k. Let A = {a1, …, a_n} be a set of nonzero vectors in N^m; each vector a_i = (a_i1, ... , a_im) corresponds to a monomial t^ai = t^ai1 …t^aim m in k[t]. The kernel of the homomorphism of k-algebras f: k [x] ->k[t]; x_i -> t^ai 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  x^a Îk[x] as a₁a₁ + …+a_na_n ÎN^m, and says that a polynomial fÎk [x] is A-homogeneous if its monomials have the same A-degree, then I_A is generated by A-homogeneous binomials.

The ideal IA is a complete intersection if there exists a system of A-homogeneous binomials g_1,…, g_s such that I_A= (g₁,…, g_s) , where s = n– rk (ZA).

If n> m and A = {d₁e_1,…, d_me_m; a_m+_1,…, a_n}, where {e₁, …, e_m} is the canonical basis of Z^m, the toric ideal I_A 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 I_A is a complete intersection without having to compute a minimal system of A-homogeneous generators of I_A. This is a joint work with Isabel Bermejo.