The quadratic sum problem for symplectic pairs

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. (c) 2023 Elsevier Inc. All rights reserved.