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- W4301935468 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 $nlfloor sqrt{n+1}rfloor$ are required to define the null cone. We generalize our results to matrix invariants of $m$-tuples of $ptimes q$ 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." @default.
- W4301935468 created "2022-10-06" @default.
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- W4301935468 date "2015-12-10" @default.
- W4301935468 modified "2023-09-26" @default.
- W4301935468 title "Polynomial degree bounds for matrix semi-invariants" @default.
- W4301935468 doi "https://doi.org/10.48550/arxiv.1512.03393" @default.
- W4301935468 hasPublicationYear "2015" @default.
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