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On the arithmetic complexity of computing Gröbner bases of comaximal determinantal ideals

Sriram Gopalakrishnan

Abstract

Let $M$ be an $n\times n$ matrix of homogeneous linear forms over a field $\Bbbk$. If the ideal $\mathcal{I}_{n-2}(M)$ generated by minors of size $n-1$ is Cohen-Macaulay, then the Gulliksen-Negård complex is a free resolution of $\mathcal{I}_{n-2}(M)$. It has recently been shown that by taking into account the syzygy modules for $\mathcal{I}_{n-2}(M)$ which can be obtained from this complex, one can derive a refined signature-based Gröbner basis algorithm DetGB which avoids reductions to zero when computing a grevlex Gröbner basis for $\mathcal{I}_{n-2}(M)$. In this paper, we establish sharp complexity bounds on DetGB. To accomplish this, we prove several results on the sizes of reduced grevlex Gröbner bases of reverse lexicographic ideals, thanks to which we obtain two main complexity results which rely on conjectures similar to that of Fröberg. The first one states that, in the zero-dimensional case, the size of the reduced grevlex Gröbner basis of $\mathcal{I}_{n-2}(M)$ is bounded from below by $n^{6}$ asymptotically. The second, also in the zero-dimensional case, states that the complexity of DetGB is bounded from above by $n^{2ω+3}$ asymptotically, where $2\leω\le 3$ is any complexity exponent for matrix multiplication over $\Bbbk$.

On the arithmetic complexity of computing Gröbner bases of comaximal determinantal ideals

Abstract

Let be an matrix of homogeneous linear forms over a field . If the ideal generated by minors of size is Cohen-Macaulay, then the Gulliksen-Negård complex is a free resolution of . It has recently been shown that by taking into account the syzygy modules for which can be obtained from this complex, one can derive a refined signature-based Gröbner basis algorithm DetGB which avoids reductions to zero when computing a grevlex Gröbner basis for . In this paper, we establish sharp complexity bounds on DetGB. To accomplish this, we prove several results on the sizes of reduced grevlex Gröbner bases of reverse lexicographic ideals, thanks to which we obtain two main complexity results which rely on conjectures similar to that of Fröberg. The first one states that, in the zero-dimensional case, the size of the reduced grevlex Gröbner basis of is bounded from below by asymptotically. The second, also in the zero-dimensional case, states that the complexity of DetGB is bounded from above by asymptotically, where is any complexity exponent for matrix multiplication over .
Paper Structure (22 sections, 25 theorems, 67 equations, 1 table, 2 algorithms)

This paper contains 22 sections, 25 theorems, 67 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gröbner bases
  4. Macaulay matrices
  5. Hilbert series
  6. Free resolutions
  7. Genericity
  8. The Hilbert series of determinantal ideals
  9. The complex of Gulliksen and Negård
  10. An explicit Hilbert series
  11. Reverse lexicographic ideals
  12. The grevlex staircase of a homogeneous reverse lexicographic ideal
  13. Reverse lexicographic determinantal ideals
  14. Determinantal ideals and the Lefschetz properties
  15. A signature-based Gröbner basis algorithm for I[n-2](M)
  16. ...and 7 more sections

Key Result

Theorem 2.4

Let $\mathbf{f}=(\mathbf{f}_1,\dots,\mathbf{f}_s)\subseteq\mathcal{R}^{m}$ be homogeneous elements. Then the rows of $\widetilde{\mathcal{M}}_{d, \succ}(\mathbf{f})$, form the elements of degree $d$ of a $\succ_{\mathrm{TOP}}$-Gröbner basis for the module generated by $\mathbf{f}$.

Theorems & Definitions (57)

  • Definition 2.1: Gröbner basis, CoxLittleOShea2005
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Definition 2.5
  • Proposition 2.6: BrunsHerzog1998
  • Theorem 2.7: Hilbert's syzygy theorem, Eisenbud1995
  • Remark 2.8
  • Corollary 2.9
  • ...and 47 more