Alternative equations for generalized polynomials on plane curves

We study the Pexiderized alternative equation (PAE): \(f(x)g(y) = 0\) for all \((x,y) \in S\), where \(f,g:{\mathbb {R}}\rightarrow {\mathbb {R}}\) are generalized polynomials and S is a plane curve. This extends the study of the alternative equation (AE): \(f(x)f(y) = 0\) for \((x,y) \in S\), where \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is an additive function or other generalized polynomial. The main question about (AE) is whether \(f=0\) is the unique solution, and for (PAE) whether it implies that \(f=0\) or \(g=0\). In the case of (AE) it is known that \(f=0\) is the unique additive solution when S is a circle centered at the origin, a curve with polynomial parametrization, or a certain form of hyperbola. Moreover some results are known for (AE) when f is assumed to be a generalized polynomial. Our findings generalize and extend those results to (PAE) and to other plane curves. As a consequence we also gain some new results about (AE).