Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Zero sets of univariate polynomials
HTML articles powered by AMS MathViewer

by Robert S. Lubarsky and Fred Richman PDF
Trans. Amer. Math. Soc. 362 (2010), 6619-6632 Request permission

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.
References
  • Errett Bishop, Foundations of constructive analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. MR 0221878
  • Thierry Coquand and Bas Spitters, Formal topology and constructive mathematics: the Gelfand and Stone-Yosida representation theorems, J.UCS 11 (2005), no. 12, 1932–1944. MR 2209804
  • S. Losinsky, Sur le procédé d’interpolation de Fejér, C. R. (Doklady) Acad. Sci. URSS (N.S.) 24 (1939), 318–321 (French). MR 0002001
  • P. Erdös, On the distribution of normal point groups, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 294–297. MR 2000, DOI 10.1073/pnas.26.4.294
  • Wim B. G. Ruitenburg, Constructing roots of polynomials over the complex numbers, Computational aspects of Lie group representations and related topics (Amsterdam, 1990) CWI Tract, vol. 84, Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1991, pp. 107–128. MR 1120034
Similar Articles
  • 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
  • Received by editor(s): February 12, 2009
  • Received by editor(s) in revised form: April 16, 2009
  • Published electronically: August 3, 2010
  • © Copyright 2010 American Mathematical Society
  • 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
  • MathSciNet review: 2678988