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

Full-text PDF

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.

**1.**Bishop, Errett, Foundations of constructive analysis, McGraw-Hill, 1967. MR**0221878 (36:4930)****2.**Coquand, Thierry and Bas Spitters, Formal topology and constructive mathematics: The Gelfand and Stone-Yosida representation theorems,*Journal of Universal Computer Science*,**11**(2005) 1932-1944 MR**2209804 (2006m:03097)****3.**Richman, Fred, The fundamental theorem of algebra: A constructive development without choice,*Pacific Journal of Mathematics*,**196**(2000), 213-230. MR**2001k:03141****4.**-, Generalized real numbers in constructive mathematics,*Indagationes Mathematicae*,**9**(1998), 595-606. MR**2000e:03172****5.**Ruitenburg, Wim B. G., Constructing roots of polynomials over the complex numbers.*Computational aspects of Lie group representations and related topics*(Amsterdam, 1990), 107-128, CWI Tract,**84**, Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1991. MR**1120034 (92g:03085)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
03F65,
13A99

Retrieve articles in all journals with MSC (2010): 03F65, 13A99

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