Ideals of Spaces of Degenerate Matrices

TL;DR

This paper investigates the algebraic structure of the variety of tuples of matrices where all linear combinations are singular, revealing that the ideal of equations defining this variety is not always generated by determinants, especially for larger m.

Contribution

It determines the vanishing ideal of the space of singular matrix tuples for 2x2 matrices and shows additional equations are needed for larger matrices, answering a question about the radicality of the ideal.

Findings

The ideal for 2x2 matrices is explicitly characterized.

For m ≥ n^2 - n + 1, additional equations beyond determinants are necessary.

The results involve classical algebraic geometry, representation theory, and computational algebra techniques.

Abstract

The variety $ \mathrm{Sing}_{n, m} $ consists of all tuples $ X = (X_1,\ldots, X_m) $ of $ n\times n $ matrices such that every linear combination of $ X_1,\ldots, X_m $ is singular. Equivalently, $X\in\mathrm{Sing}_{n,m}$ if and only if $\det(\lambda_1 X_1 + \ldots + \lambda_m X_m) = 0$ for all $ \lambda_1,\ldots, \lambda_m\in \mathbb Q $. Makam and Wigderson asked whether the ideal generated by these equations is always radical, that is, if any polynomial identity that is valid on $ \mathrm{Sing}_{n, m} $ lies in the ideal generated by the polynomials $\det(\lambda_1 X_1 + \ldots + \lambda_m X_m)$. We answer this question in the negative by determining the vanishing ideal of $ \mathrm{Sing}_{2, m} $ for all $ m\in \mathbb N $. Our results exhibit that there are additional equations arising from the tensor structure of $ X $. More generally, for any $ n $ and $ m\ge n^2 - n + 1 $, we prove there are equations vanishing on $ \mathrm{Sing}_{n, m} $ that are not in the ideal generated by polynomials of type $\det(\lambda_1 X_1 + \ldots + \lambda_m X_m)$. Our methods are based on classical results about Fano schemes, representation theory and Gr\"obner bases.