We consider two group actions on -tuples of matrices with entries in the field . The first is simultaneous conjugation by and the second is the left-right action of . Let be the algebraic closure of the field . Recently, a polynomial time algorithm was found to decide whether lies in the Zariski closure of the -orbit of a given -tuple by Garg, Gurvits, Oliveira and Wigderson for the base field . 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 -tuples intersect. For the conjugation action of a polynomial time algorithm for orbit closure separation was given by Forbes and Shpilka in characteristic . Here, we give a polynomial time algorithm for the orbit closure separation problem for both the conjugation action of and the left-right action of in arbitrary characteristic. We also improve the known bounds for the degree of separating invariants in these cases.
"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