Separating invariants over finite fields (Q2664611)

Separating invariants over finite fields are studied in the paper. Let \(V\) be a finite dimensional vector space over the finite field \(\mathbb{F}_q\), and \(G\) a group of linear transformations of \(V\). A subset \(S\subseteq \mathbb{F}_q[V]^G\) of polynomial invariants is \textit{separating} if whenever for \(v,w\in V\) there is an \(f\in \mathbb{F}_q[V]^G\) with \(f(v)\neq f(w)\), then there exists an \(h\in S\) such that \(h(v)\neq h(w)\). The authors show that there exists a separating set for \(\mathbb{F}_q[V]^G\) of size \(\gamma:=\lceil\log_q(k)\rceil\), where \(k\) is the number of \(G\)-orbits in \(V\). This is the minimal possible size of a separating set. Moreover, they construct a separating set of size \(\gamma\) consisting of polynomials of degree at most \(|G|\dim(V)(q-1)\). This degree bound is improved in certain special cases. For example, in the non-modular case there exists a separating set of size \(\gamma\) consisting of elements of degree at most \(\dim(V)(q-1)\). The study of separating invariants over finite fields has relevance for discrete mathematics. This is demonstrated by the detailed study of a permutation representation related to the graph isomorphism problem. In the final section working over the \(2\)-element field, a minimal (with respect to inclusion) separating set of multisymmetric polynomials is found.