Alternative equations for generalized polynomials

Abstract

The alternative functional equation f(x) g(y) = 0 for (x,y) belonging to a subset D of the Cartesian product of the domains of the functions f and g is investigated under the assumptions that f and g are generalized polynomials and D is a particular conic section or it has a positive product measure or it is a second category Baire set. In several cases it is proved that f or g equals zero identically.