Polynomial degree bounds for matrix semi-invariants
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.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2015
- DOI:
- 10.48550/arXiv.1512.03393
- arXiv:
- arXiv:1512.03393
- Bibcode:
- 2015arXiv151203393D
- Keywords:
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- Mathematics - Representation Theory;
- Computer Science - Computational Complexity;
- 13A50 (Primary);
- 14L24;
- 16G20 (Secondary)
- E-Print:
- 16 pages