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The algebra of ${\rm GL}_n$-invariants of $m$-tuples of $n\times n$ matriceswith respect to the action by simultaneous conjugation is a classical topic incase of infinite base field. On the other hand, in case of a finite fieldgenerators 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 ${\rmGL}_2$-orbits on $m$-tuples of $2\times 2$ nilpotent matrices for all $m>0$. Asa consequence, we obtained a minimal separating set for the algebra of ${\rmGL}_2$-invariant polynomial functions of $m$-tuples of $2\times 2$ nilpotentmatrices. We also described the least possible number of elements of aseparating set for an algebra of invariant polynomial functions over a finitefield.

Separating invariants of several nilpotent 2x2 matrices