Zero sets of univariate polynomials

Authors:
Robert S. Lubarsky and Fred Richman

Journal:
Trans. Amer. Math. Soc. **362** (2010), 6619-6632

MSC (2010):
Primary 03F65, 13A99

DOI:
https://doi.org/10.1090/S0002-9947-2010-05086-X

Published electronically:
August 3, 2010

MathSciNet review:
2678988

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Abstract | References | Similar Articles | Additional Information

Abstract: Let 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 . 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 . 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.

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Additional Information

**Robert S. Lubarsky**

Affiliation:
Department of Mathematics, Florida Atlantic University, Boca Raton, Florida 33431-0991

Email:
Robert.Lubarsky@comcast.net

**Fred Richman**

Affiliation:
Department of Mathematics, Florida Atlantic University, Boca Raton, Florida 33431-0991

Email:
richman@fau.edu

DOI:
https://doi.org/10.1090/S0002-9947-2010-05086-X

Received by editor(s):
February 12, 2009

Received by editor(s) in revised form:
April 16, 2009

Published electronically:
August 3, 2010

Article copyright:
© Copyright 2010
American Mathematical Society