Polynomial degree bounds for matrix semiinvariants
Abstract
We study the leftright 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 semiinvariants 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 blowups of matrix spaces. We will discuss several applications to algebraic complexity theory, such as a deterministic polynomial time algorithm for noncommutative rational identity testing, and the existence of small divisionfree formulas for noncommutative polynomials.
 Publication:

arXiv eprints
 Pub Date:
 December 2015
 arXiv:
 arXiv:1512.03393
 Bibcode:
 2015arXiv151203393D
 Keywords:

 Mathematics  Representation Theory;
 Computer Science  Computational Complexity;
 13A50 (Primary);
 14L24;
 16G20 (Secondary)
 EPrint:
 16 pages