Polynomial degree bounds for matrix semi-invariants

TL;DR

This paper establishes polynomial degree bounds for matrix semi-invariants, showing how they generate the invariant ring and defining the null cone, with implications for algebraic complexity theory.

Contribution

It provides explicit degree bounds for matrix semi-invariants that generate the invariant ring and define the null cone, extending to quivers and applying new techniques.

Findings

Invariants of degree n^2 - n define the null cone.

Invariants of degree ≤ n^6 generate the ring of invariants in characteristic zero.

Higher degree invariants are required for large m to define the null cone.

Abstract

We study the left-right action of $\operatorname{SL}_n \times \operatorname{SL}_n$ on $m$-tuples of $n \times n$ matrices with entries in an infinite field $K$. We show that invariants of degree $n^2- n$ define the null cone. Consequently, invariants of degree $\leq n^6$ generate the ring of invariants if $\operatorname{char}(K)=0$. We also prove that for $m \gg 0$, invariants of degree at least $n\lfloor \sqrt{n+1}\rfloor$ are required to define the null cone. We generalize our results to matrix invariants of $m$-tuples of $p\times q$ matrices, and to rings of semi-invariants for quivers. For the proofs, we use new techniques such as the regularity lemma by Ivanyos, Qiao and Subrahmanyam, and the concavity property of the tensor blow-ups of matrix spaces. We will discuss several applications to algebraic complexity theory, such as a deterministic polynomial time algorithm for non-commutative rational identity testing, and the existence of small division-free formulas for non-commutative polynomials.