Algorithms for orbit closure separation for invariants and semi-invariants of matrices

Derksen, Harm; Makam, Visu

doi:10.2140/ant.2020.14.2791

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Algorithms for orbit closure separation for invariants and semi-invariants of matrices

Harm Derksen and Visu Makam

Vol. 14 (2020), No. 10, 2791–2813

DOI: 10.2140/ant.2020.14.2791

Abstract

We consider two group actions on $m$ -tuples of $n \times n$ matrices with entries in the field $K$ . The first is simultaneous conjugation by ${GL}_{n}$ and the second is the left-right action of ${SL}_{n} \times {SL}_{n}$ . Let $\bar{K}$ be the algebraic closure of the field $K$ . Recently, a polynomial time algorithm was found to decide whether $0$ lies in the Zariski closure of the ${SL}_{n} (\bar{K}) \times {SL}_{n} (\bar{K})$ -orbit of a given $m$ -tuple by Garg, Gurvits, Oliveira and Wigderson for the base field $K = ℚ$ . An algorithm that also works for finite fields of large enough cardinality was given by Ivanyos, Qiao and Subrahmanyam. A more general problem is the orbit closure separation problem that asks whether the orbit closures of two given $m$ -tuples intersect. For the conjugation action of ${GL}_{n} (\bar{K})$ a polynomial time algorithm for orbit closure separation was given by Forbes and Shpilka in characteristic $0$ . Here, we give a polynomial time algorithm for the orbit closure separation problem for both the conjugation action of ${GL}_{n} (\bar{K})$ and the left-right action of ${SL}_{n} (\bar{K}) \times {SL}_{n} (\bar{K})$ in arbitrary characteristic. We also improve the known bounds for the degree of separating invariants in these cases.

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Keywords

orbit closure intersection, null cone, matrix semi-invariants, matrix invariants, separating invariants

Mathematical Subject Classification 2010

Primary: 13A50

Secondary: 14L24, 68W30

Milestones

Received: 24 October 2019

Revised: 15 March 2020

Accepted: 20 June 2020

Published: 19 November 2020

Authors

Harm Derksen
Department of Mathematics University of Michigan Ann Arbor, MI United States

Visu Makam
School of Mathematics Institute for Advanced Study Princeton, NJ United States

Vol. 14, No. 10, 2020