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Regularity and algebraic properties of certain lattice ideals

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Abstract

We study the regularity and the algebraic properties of certain lattice ideals. We establish a map \(I \mapsto \tilde I\) between the family of graded lattice ideals in an ℕ-graded polynomial ring over a field K and the family of graded lattice ideals in a polynomial ring with the standard grading. This map is shown to preserve the complete intersection property and the regularity of I but not the degree. We relate the Hilbert series and the generators of I and \(\tilde I\). If dim(I) = 1, we relate the degrees of I and \(\tilde I\). It is shown that the regularity of certain lattice ideals is additive in a certain sense. Then, we give some applications. For finite fields, we give a formula for the regularity of the vanishing ideal of a degenerate torus in terms of the Frobenius number of a semigroup. We construct vanishing ideals, over finite fields,with prescribed regularity and degree of a certain type. Let X be a subset of a projective space over a field K. It is shown that the vanishing ideal of X is a lattice ideal of dimension 1 if and only if X is a finite subgroup of a projective torus. For finite fields, it is shown that X is a subgroup of a projective torus if and only if X is parameterized by monomials. We express the regularity of the vanishing ideal over a bipartite graph in terms of the regularities of the vanishing ideals of the blocks of the graph.

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Author information

Authors and Affiliations

  1. CMUC, Department of Mathematics, University of Coimbra, 3001-454, Coimbra, Portugal

    Jorge Neves

  2. Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Avenida Rovisco Pais, 1, 1049-001, Lisboa, Portugal

    Maria Vaz Pinto

  3. Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14-740, 07000, Mexico City, D.F, Mexico

    Rafael H. Villarreal

Authors
  1. Jorge Neves
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  2. Maria Vaz Pinto
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  3. Rafael H. Villarreal
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Corresponding author

Correspondence to Rafael H. Villarreal.

Additional information

Dedicated to Aron Simis on the occasion of his 70th birthday.

The first author was partially funded by CMUC and FCT (Portugal), through European program COMPETE/FEDER and project PTDC/MAT/111332/2009, and by a research grant from Santander Totta Bank (Portugal).

The second author is amember of the Center forMathematicalAnalysis,Geometry, andDynamical Systems, Departamento de Matematica, Instituto Superior Tecnico, 1049-001 Lisboa, Portugal. The third author was partially supported by SNI.

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Neves, J., Pinto, M.V. & Villarreal, R.H. Regularity and algebraic properties of certain lattice ideals. Bull Braz Math Soc, New Series 45, 777–806 (2014). https://doi.org/10.1007/s00574-014-0075-5

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  • DOI: https://doi.org/10.1007/s00574-014-0075-5

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