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[t_{1}, …, 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 a_{1}a_{1 }+ …+a_{n}a_{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}= (g_{1},…, g_{s}) , where s = n– rk (ZA).

If n> m and A = {d_{1}e_{1,}…, d_{m}e_{m}; a_{m}+_{1,…, }a_{n}}, where {e_{1},
…, 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.