On <i>m</i>-tuples of nilpotent 2 × 2 matrices over an arbitrary field

The algebra of [Formula: see text]-invariants of m-tuples of [Formula: see text] 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 are not known even for a pair of [Formula: see text] matrices. Working over an arbitrary field we classified all [Formula: see text]-orbits on m-tuples of [Formula: see text] nilpotent matrices for all [Formula: see text]. As a consequence, we obtained a minimal separating set for the algebra of [Formula: see text]-invariant polynomial functions of m-tuples of [Formula: see text] 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.