Zero sets of univariate polynomials

Abstract:

Let $L$ be the zero set of a nonconstant monic polynomial with complex coefficients. In the context of constructive mathematics without countable choice, it may not be possible to construct an element of $L$. In this paper we introduce a notion of distance from a point to a subset, more general than the usual one, that allows us to measure distances to subsets such as $L$. To verify the correctness of this notion, we show that the zero set of a polynomial cannot be empty—a weak fundamental theorem of algebra. We also show that the zero sets of two polynomials are a positive distance from each other if and only if the polynomials are comaximal. Finally, the zero set of a polynomial is used to construct a separable Riesz space, in which every element is normable, that has no Riesz homomorphism into the real numbers.