Zero Sets of Univariate Polynomials

TL;DR

This paper introduces a new notion of distance to analyze zero sets of polynomials within constructive mathematics, proving fundamental properties and constructing a specific Riesz space with unique features.

Contribution

It defines a generalized distance concept to zero sets, proves a weak fundamental theorem of algebra, and constructs a Riesz space with no Riesz homomorphism into reals.

Findings

Zero set of a polynomial cannot be empty.

Zero sets of two polynomials are positively distant iff they are coprime.

Constructed Riesz space has no Riesz homomorphism into real numbers.

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 like $L$. 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.