In this paper, we use the tools of Gr\"{ o}bner bases and combinatorial secant varieties to study the determinantal ideals $I_t$ of the extended Hankel matrices. Denote by $c$-chain a sequence $a_1,\...,a_k$ 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 $I_t^{ (s)}$ and we compute the primary decomposition of the product $I_{ t_1}\... I_{ t_k}$ of the determinantal ideals. Passing through the initial ideals and algebras we prove that the product $I_{ t_1}\... I_{ t_k}$ has a linear resolution and the multi-homogeneous Rees algebra $\Rees(I_{ t_1},\...,I_{ t_k})$ is defined by a Gr\"obner basis of quadrics.
Categories: math.AC math.CO