Restriction of a quadratic form over a nite eld to a nondegenerate affine quadric hypersurface

Abstract

Let h;h M : F n q ! F q be quadratic forms with h not degenerate. Fix k 2 F q and set C n ( k;h ) F q := f h ( x 1 ;:::;x n ) = k gF n q . We compute (in many cases) the image of h M j C n ( k;h ) F q . This question is related to a question on the numerical range of matrices over a nite eld.