A Generalization of Walsh’s Two-Circle Theorem to Vector Spaces

In this article we investigate the location of the null sets of generalized pseudo-derivatives of the product or quotient. of abstract polynomials. Some special cases of this general problem were studied by Walsh as geometry of polynomials in the complex plane. One of our two results deduces an improved version of the first part of the two-circle theorem due to Walsh on the quotient of two polynomials and the other furnishes an improved version of the two-circle theorem on the product of two polynomials.