Vol. 14, No. 10, 2020

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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

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.

orbit closure intersection, null cone, matrix semi-invariants, matrix invariants, separating invariants
Mathematical Subject Classification 2010
Primary: 13A50
Secondary: 14L24, 68W30
Received: 24 October 2019
Revised: 15 March 2020
Accepted: 20 June 2020
Published: 19 November 2020
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