Polynomial degree bounds for matrix semi-invariants

Abstract

We study the left–right action of SLn×SLn on m-tuples of n×n matrices with entries in an infinite field K. We show that invariants of degree n2−n define the null cone. Consequently, invariants of degree ≤n6 generate the ring of invariants if char(K)=0. We also prove that for m≫0, invariants of degree at least n⌊n+1⌋ are required to define the null cone. We generalize our results to matrix invariants of m-tuples of p×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.