Polynomial degree bounds for matrix semi-invariants

Abstract

We study the left-right action of ${\mathrm{SL}}_n\times {\mathrm{SL}}_n$ 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 $\le n^6$ generate the ring of invariants if $\mathrm{char}(K)=0$. 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.

16 pages