On m-tuples of nilpotent 2x2 matrices over an arbitrary field

The algebra of ${\rm GL}_n$-invariants of $m$-tuples of $n\times n$ matrices with respect to the action by simultaneous conjugation is a classical topic in case of infinite base field. On the other hand, in case of a finite field generators of polynomial invariants even in case of a pair of $2\times 2$ matrices are not known. Working over an arbitrary field we classified all ${\rm GL}_2$-orbits on $m$-tuples of $2\times 2$ nilpotent matrices for all $m>0$. As a consequence, we obtained a minimal separating set for the algebra of ${\rm GL}_2$-invariant polynomial functions of $m$-tuples of $2\times 2$ nilpotent matrices. We also described the least possible number of elements of a separating set for an algebra of invariant polynomial functions over a finite field.