Name
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. |
Year | Venue | DocType |
|---|---|---|
2015 | CoRR | Journal |
Volume | Citations | PageRank |
abs/1512.03393 | 11 | 0.90 |
References | Authors | |
4 | 2 | |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Harm Derksen | 1 | 151 | 15.00 |
| Visu Makam | 2 | 11 | 2.59 |