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The Gauss Algebra of squarefree Veronese algebras

Somayeh Bandari, Raheleh Jafari

TL;DR

This work determines the Gauss algebra $G(A)$ for squarefree Veronese algebras generated in degree $3$, focusing on dimensions $d\le7$. For $d=3,4$ the Gauss algebra is a one-dimensional polynomial ring, while for $d=5,6,7$ it has a concrete monomial description $K[Mon^*_S(4,2d)\setminus E_d]$ and the corresponding monomial ideal is polymatroidal, ensuring normality and Cohen-Macaulayness. The authors provide explicit generators in the cases $d=5,6,7$ and prove the polymatroidal structure, supporting a broader conjecture that the same description holds for all $d\ge5$; higher dimensions $d\ge8$ are left as an open question. These results connect Gauss algebras to discrete polymatroid theory and yield structural, homological properties for these algebras. Data is not applicable since no new data were collected.

Abstract

We investigate the Gauss algebra for squarefree Veronese algebras generated in degree $3$. For small dimensions not exceeding $7$, we determine the Gauss algebra by specifying its generators and show in particular that it is normal and Cohen-Macaulay.

The Gauss Algebra of squarefree Veronese algebras

TL;DR

This work determines the Gauss algebra for squarefree Veronese algebras generated in degree , focusing on dimensions . For the Gauss algebra is a one-dimensional polynomial ring, while for it has a concrete monomial description and the corresponding monomial ideal is polymatroidal, ensuring normality and Cohen-Macaulayness. The authors provide explicit generators in the cases and prove the polymatroidal structure, supporting a broader conjecture that the same description holds for all ; higher dimensions are left as an open question. These results connect Gauss algebras to discrete polymatroid theory and yield structural, homological properties for these algebras. Data is not applicable since no new data were collected.

Abstract

We investigate the Gauss algebra for squarefree Veronese algebras generated in degree . For small dimensions not exceeding , we determine the Gauss algebra by specifying its generators and show in particular that it is normal and Cohen-Macaulay.
Paper Structure (3 sections, 10 theorems, 10 equations)

This paper contains 3 sections, 10 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. The Gauss algebra of squarefree Veronese algebras
  3. Data Availability

Key Result

Proposition 2.1

HJN Let $A={\mathbb K}[{\mathbf g}]$ be a homogeneous monomial algebra in $S$ with generators ${\mathbf g}=g_1,\ldots,g_n$. Then the Gauss algebra of $A$ is determined as

Theorems & Definitions (20)

  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 10 more