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arXiv · 1812.06529

Generalized minimum distance functions and algebraic invariants of Geramita ideals

Susan M. Cooper·Alexandra Seceleanu·Stefan O. Tohaneanu·Maria Vaz Pinto·Rafael H. Villarreal
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Abstract

Motivated by notions from coding theory, we study the generalized minimum distance (GMD) function $δ_I(d,r)$ of a graded ideal $I$ in a polynomial ring over an arbitrary field using commutative algebraic methods. It is shown that $δ_I$ is non-decreasing as a function of $r$ and non-increasing as a function of $d$. For vanishing ideals over finite fields, we show that $δ_I$ is strictly decreasing as a function of $d$ until it stabilizes. We also study algebraic invariants of Geramita ideals. Those ideals are graded, unmixed, $1$-dimensional and their associated primes are generated by linear forms. We also examine GMD functions of complete intersections and show some special cases of two conjectures of Tohăneanu--Van Tuyl and Eisenbud-Green-Harris.

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Susan M. Cooper, Alexandra Seceleanu, Stefan O. Tohaneanu, Maria Vaz Pinto, Rafael H. Villarreal. 2019-09-12. Generalized minimum distance functions and algebraic invariants of Geramita ideals. https://arxiv.org/abs/1812.06529

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