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The determinantal ideals of extended Hankel matrices

Abstract

In this paper, we use the tools of Gr\"{o}bner bases and combinatorial secant varieties to study the determinantal ideals ItI_t of the extended Hankel matrices. Denote by cc-chain a sequence a1,.˙.,aka_1,\...,a_k with ai+c<ai+1a_i+c<a_{i+1} for all i=1,.˙.,k1i=1,\...,k-1. Using the results of cc-chain, we solve the membership problem for the symbolic powers It(s)I_t^{(s)} and we compute the primary decomposition of the product It1.˙.ItkI_{t_1}\... I_{t_k} of the determinantal ideals. Passing through the initial ideals and algebras we prove that the product It1.˙.ItkI_{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

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