The determinantal ideals of extended Hankel matrices

In this paper, we use the tools of Gröbner bases and combinatorial secant varieties to study the determinantal ideals I_t of the extended Hankel matrices. Denote by c-chain a sequence a₁,…,a_k with a_i+c<a_i+1 for all i=1,…,k−1. Using the results of c-chain, we solve the membership problem for the symbolic powers and we compute the primary decomposition of the product I_t1?I_tk of the determinantal ideals. Passing through the initial ideals and algebras we prove that the product I_t1?I_tk has a linear resolution and the multi-homogeneous Rees algebra is defined by a Gröbner basis of quadrics.