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

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 $\overline{K}$ 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 ${SL}_n\left(\overline{K}\right)\times {SL}_n\left(\overline{K}\right)$-orbit of a given $m$-tuple by Garg, Gurvits, Oliveira and Wigderson for the base field $K=\mathbb{Q}$. 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 ${GL}_n\left(\overline{K}\right)$ 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 ${GL}_n\left(\overline{K}\right)$ and the left-right action of ${SL}_n\left(\overline{K}\right)\times {SL}_n\left(\overline{K}\right)$ in arbitrary characteristic. We also improve the known bounds for the degree of separating invariants in these cases.