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Minimal generating sets of lattice ideals

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Abstract

Let \(L\subset \mathbb {Z}^n\) be a lattice and \(I_L=\langle x^{\mathbf {u}}-x^{\mathbf {v}}:\ {\mathbf {u}}-{\mathbf {v}}\in L\rangle \) be the corresponding lattice ideal in \(\Bbbk [x_1,\ldots , x_n]\), where \(\Bbbk \) is a field. In this paper we describe minimal binomial generating sets of \(I_L\) and their invariants. We use as a main tool a graph construction on equivalence classes of fibers of \(I_L\). As one application of the theory developed we characterize binomial complete intersection lattice ideals, a longstanding open problem in the case of non-positive lattices.

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Notes

  1. The notion of Markov basis was introduced in [10] to denote a generating binomial set of \(I_L\), see [10, Theorem 3.1]. Occasionally in the literature, especially in algebraic statistics, the minimal generating sets are also referred to as minimal Markov bases.

  2. Even though for positive lattices the notions of minimal generating sets and cardinality-minimal generating sets coincide, this is not true for non-positive lattices.

  3. The union of all minimal generating sets of \(I_L\) is called the universal Markov basis of \(I_L\), following [19, Definition 3.1].

  4. Let \(p_1,\ldots , p_s\) be s distinct primes and let \(a_i=(p_1\cdots p_s)/p_i\). Then \(\langle 1-(xy)^{a_1}, \ldots , 1-(xy)^{a_s}\rangle =\langle 1-xy\rangle \) and \( \{ 1-x^5, 1-(xy)^{a_1},\ldots , 1-(xy)^{a_s}\}\) is a minimal generating set of \(I_L\) of cardinality \(s+1\).

  5. There are cardinality-minimal generating sets that are not contained in the Graver basis of \(I_L\), as is the case for \(\{1-x^{2012}y^{2017}, y^4-x^{2013}y^{2022}\} \). The union of the cardinality-minimal generating sets of \(I_L\) is an infinite set.

  6. This is proved in detail in the last section.

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Acknowledgements

The authors would like to thank Ezra Miller for useful discussions on binomial fibers, back in 2009, that initiated this work. Also, we thank the anonymous referee for several helpful suggestions on the terminology of this paper and for simplifying the proofs of Propositions 2.11, 5.2 and Theorems 2.12, 2.13. This paper was partially written during the visit of the second and third author at the University of Ioannina. The third author was supported by a Romanian Grant awarded by UEFISCDI, Project Number 83 / 2010, PNII-RU code TE\(\_46/2010\), program Human Resources, “Algebraic modeling of some combinatorial objects and computational applications”.

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Authors and Affiliations

  1. Department of Mathematics, Aristotle University of Thessaloniki, 54124, Thessaloníki, Greece

    Hara Charalambous

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

    Apostolos Thoma

  3. Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, 010014, Bucharest, Romania

    Marius Vladoiu

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  1. Hara Charalambous
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  2. Apostolos Thoma
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  3. Marius Vladoiu
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Correspondence to Marius Vladoiu.

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Charalambous, H., Thoma, A. & Vladoiu, M. Minimal generating sets of lattice ideals. Collect. Math. 68, 377–400 (2017). https://doi.org/10.1007/s13348-017-0191-9

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