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On robustness and related properties on toric ideals

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Abstract

A toric ideal is called robust if its universal Gröbner basis is a minimal set of generators, and is called generalized robust if its universal Gröbner basis equals its universal Markov basis (the union of all its minimal sets of binomial generators). Robust and generalized robust toric ideals are both interesting from both a commutative algebra and an algebraic statistics perspective. However, only a few nontrivial examples of such ideals are known. In this work, we study these properties for toric ideals of both graphs and numerical semigroups. For toric ideals of graphs, we characterize combinatorially the graphs giving rise to robust and to generalized robust toric ideals generated by quadratic binomials. As a by-product, we obtain families of Koszul rings. For toric ideals of numerical semigroups, we determine that one of its initial ideals is a complete intersection if and only if the semigroup belongs to the so-called family of free numerical semigroups. Hence, we characterize all complete intersection numerical semigroups which are minimally generated by one of its Gröbner basis and, as a consequence, all the Betti numbers of the toric ideal and its corresponding initial ideal coincide. Moreover, also for numerical semigroups, we prove that the ideal is generalized robust if and only if the semigroup has a unique Betti element and that there are only trivial examples of robust ideals. We finish the paper with some open questions.

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Acknowledgements

This paper was written during the visit of the second author at the Department of Mathematics in Universidad de La Laguna (ULL). This work was partially supported by the Spanish MICINN ALCOIN (PID2019-104844GB-I00) and by the ULL funded research projects MASCA and MACACO.Computational experiments with the computer softwares CoCoA [1] and Singular [19] have helped in the elaboration of this work. We also would like to thank the anonymous referees for their positive comments and careful reviews, which helped improve the manuscript.

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  1. Instituto de Matemáticas y Aplicaciones (IMAULL), Sección de Matemáticas, Facultad de Ciencias, Universidad de La Laguna, 38200, La, Laguna, Spain

    Ignacio García-Marco

  2. Department of Mathematics, University of Ioannina, 45110, Ioannina, Greece

    Christos Tatakis

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  1. Ignacio García-Marco
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  2. Christos Tatakis
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Correspondence to Christos Tatakis.

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García-Marco, I., Tatakis, C. On robustness and related properties on toric ideals. J Algebr Comb 57, 21–52 (2023). https://doi.org/10.1007/s10801-022-01162-x

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