The quadratic sum problem for symplectic pairs

Abstract

Let (b,u) be a pair consisting of a symplectic form b on a finite-dimensional vector space V over a field F, and of a b-alternating endomorphism u of V (i.e. b(x,u(x))=0 for all x in V).Let p and q be arbitrary polynomials of degree 2 with coefficients in F. We characterize, in terms of the invariant factors of u, the condition that u splits into u1+u2 for some pair (u1,u2) of b-alternating endomorphisms such that p(u1)=q(u2)=0.