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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: August 3, 2010
MathSciNet review: 2678988
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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.


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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: http://dx.doi.org/10.1090/S0002-9947-2010-05086-X
PII: S 0002-9947(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