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2020 Algorithms for orbit closure separation for invariants and semi-invariants of matrices
Harm Derksen, Visu Makam
Algebra Number Theory 14(10): 2791-2813 (2020). DOI: 10.2140/ant.2020.14.2791

Abstract

We consider two group actions on m-tuples of n×n matrices with entries in the field K. The first is simultaneous conjugation by GLn and the second is the left-right action of SLn× SLn. Let 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 SLn(K¯)× SLn(K¯)-orbit of a given m-tuple by Garg, Gurvits, Oliveira and Wigderson for the base field K=. 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 GLn(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 GLn(K¯) and the left-right action of SLn(K¯)× SLn(K¯) in arbitrary characteristic. We also improve the known bounds for the degree of separating invariants in these cases.

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Harm Derksen. Visu Makam. "Algorithms for orbit closure separation for invariants and semi-invariants of matrices." Algebra Number Theory 14 (10) 2791 - 2813, 2020. https://doi.org/10.2140/ant.2020.14.2791

Information

Received: 24 October 2019; Revised: 15 March 2020; Accepted: 20 June 2020; Published: 2020
First available in Project Euclid: 22 December 2020

MathSciNet: MR4190419
Digital Object Identifier: 10.2140/ant.2020.14.2791

Subjects:
Primary: 13A50
Secondary: 14L24, 68W30

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.14 • No. 10 • 2020
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