Typical separating invariants

Abstract

It is shown that a trivial version of polarization is sufficient to produce separating systems of polynomial invariants: If two points in the direct sum of the G-modules W and m copies of V can be separated by polynomial invariants, then they can be separated by invariants depending only on $\leqslant 2\dim(V)$ variables of type V; when G is reductive, invariants depending only on $\leqslant \dim(V)+1$ variables suffice. A similar result is valid for rational invariants. Explicit bounds on the number of type V variables in a complete system of typical separating invariants are given for the binary polyhedral groups, and this is applied to the invariant theory of binary forms.