Gröbner Bases of Certain Determinantal Ideals

Abstract: \font\msbm=msbm10 \def\bC{\hbox{\msbm C}} Let $\mu_1<\cdots<\mu_k$ be non-negative integers. We prove that the ideal of the locus of points $(x_1,\ldots,x_n)$ in $\bC^n$ for which the $n\times k$ matrix $(x_i^{\mu_j})$ has rank smaller than $k$ is generated by the determinants of the $k\times k$ minors of this matrix, if $\mu_1=0$ or $1$. Moreover, these generators form a universal Gröbner basis. Special cases of our result apply for ideals arising in the study of chromatic numbers of graphs or identities of matrices, and ideals of truncations of hyperplane arrangements related to pseudo-reflection groups.

Keywords: determinantal ideal, Gröbner basis, degree, Hilbert polynomial, subspace arrangement, reflection group

Classification (MSC91): 13C40; 13P10, 14M12

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