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.
