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Mathematics > Commutative Algebra

arXiv:1812.06529 (math)
[Submitted on 16 Dec 2018 (v1), last revised 12 Sep 2019 (this version, v2)]

Title:Generalized minimum distance functions and algebraic invariants of Geramita ideals

Authors: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 $\delta_I(d,r)$ of a graded ideal $I$ in a polynomial ring over an arbitrary field using commutative algebraic methods. It is shown that $\delta_I$ 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 $\delta_I$ 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.
Subjects: Commutative Algebra (math.AC); Information Theory (cs.IT); Algebraic Geometry (math.AG); Combinatorics (math.CO)
MSC classes: 13P25, 14G50, 94B27, 11T71
Cite as: arXiv:1812.06529 [math.AC]
  (or arXiv:1812.06529v2 [math.AC] for this version)
  https://doi.org/10.48550/arXiv.1812.06529
Journal reference: Adv. in Appl. Math. 112 (2020), 101940

Submission history

From: Rafael Villarreal H [view email]
[v1] Sun, 16 Dec 2018 20:18:12 UTC (33 KB)
[v2] Thu, 12 Sep 2019 01:45:04 UTC (33 KB)
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