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Sagbi bases, defining ideals and algebra of minors

Winfried Bruns, Aldo Conca, Francesca Lembo

TL;DR

The paper extends SAGBI techniques to compute defining ideals of subalgebras generated by minors and provides a complete classification for when minor-generated algebras admit a SAGBI basis, using Newton polytopes and Cartwright–Sturmfels theory. It shows that the SAGBI approach can outperform elimination in select cases and delivers universal SAGBI bases for several small but important cases ($A_2(3,3)$, $G(3,6)$, $G(3,7)$). The results illuminate when minors form a SAGBI basis (or not) and demonstrate concrete computational tools (via sagbiNormaliz, SagbiGrass, Normaliz) that aid in toric degenerations and the study of determinantal varieties. The work contributes both theoretical classifications and practical algorithms for handling algebras of minors and their initial algebras.

Abstract

This paper extends the article of the Bruns and Conca on SAGBI bases and their computation (J. Symb. Comput. 120 (2024)) in two directions. (i) We describe the extension of the Singular library sagbiNormaliz.sing to the computation of defining ideals of subalgebras of polynomial rings. (ii) We give a complete classification of the algebras of minors for which the generating set is a SAGBI basis with respect to a suitable monomial order and we identify universal SAGBI basis in three cases. The investigation is illustrated by several examples.

Sagbi bases, defining ideals and algebra of minors

TL;DR

The paper extends SAGBI techniques to compute defining ideals of subalgebras generated by minors and provides a complete classification for when minor-generated algebras admit a SAGBI basis, using Newton polytopes and Cartwright–Sturmfels theory. It shows that the SAGBI approach can outperform elimination in select cases and delivers universal SAGBI bases for several small but important cases (, , ). The results illuminate when minors form a SAGBI basis (or not) and demonstrate concrete computational tools (via sagbiNormaliz, SagbiGrass, Normaliz) that aid in toric degenerations and the study of determinantal varieties. The work contributes both theoretical classifications and practical algorithms for handling algebras of minors and their initial algebras.

Abstract

This paper extends the article of the Bruns and Conca on SAGBI bases and their computation (J. Symb. Comput. 120 (2024)) in two directions. (i) We describe the extension of the Singular library sagbiNormaliz.sing to the computation of defining ideals of subalgebras of polynomial rings. (ii) We give a complete classification of the algebras of minors for which the generating set is a SAGBI basis with respect to a suitable monomial order and we identify universal SAGBI basis in three cases. The investigation is illustrated by several examples.

Paper Structure

This paper contains 8 sections, 14 theorems, 74 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Basics of Sagbi bases
  3. Computing the defining ideal
  4. Computational data
  5. Coherent matchings, Newton polytopes and computational tools
  6. Dimension of coherent matchings for $G(m,n)$
  7. Are the minors a Sagbi basis?
  8. Universal Sagbi bases for $A_2(3,3)$, $G(3,6)$ and $G(3,7)$

Key Result

Lemma 2.1

Let $R$ be a polynomial ring, endowed with a monomial order and the standard grading. Let $A$ be a finitely generated graded subalgebra. Furthermore let ${\mathcal{F}}$ be a family of polynomials in $A$ and $B=K[\operatorname{in}({\mathcal{F}})]$ the subalgebra generated by the monomials $\operatorn

Theorems & Definitions (38)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Definition 2.3
  • Proposition 2.4
  • Remark 3.1
  • Remark 4.1
  • Remark 5.1
  • Remark 5.2
  • Theorem 6.1
  • ...and 28 more