IJPAM: Volume 86, No. 1 (2013)

COMPUTABLE IMPLEMENTATION OF
``FUNDAMENTAL THEOREM OF ALGEBRA"

Jon A. Sjogren1, Xinkai Li2, Mu Zhao2, Chao Lu2 1Mail Stop 715, 2308 Mt. Vernon Ave.
Alexandria VA 22301, USA
2Department of Computer & Information Sciences
Towson University
8000 York Rd, Towson, MD 21252, USA


Abstract. The Fundamental Theorem of Algebra (FTA) has been studied for more than 300 years: more or less satisfactory proofs of FTA emerged in the 18th and 19th centuries. Proofs denoted as `algebraic' or `elementary' derived from the axioms defining a Real-Closed Field (RCF). A proof is given that brings up-to-date work of Gauss (1816) and P. Gordan (1879). It does not refer explicitly to the complex numbers but instead works with auxiliary polynomials in two variables. We report that computer software has been developed to effect symbolic calculation in the context of exact arithmetic. Some examples show how these routines apply to the algebra of symmetric multinomial forms used in Laplace's proof (1795) of FTA, as well as to the theory of Sylvester forms and the Beezoutian formulation of the resultant.

Received: April 4, 2013

AMS Subject Classification: 08-02

Key Words and Phrases: Fundamental Theorem Algebra (FTA), Real Closed Field (RCF), symmetric polynomial, symbolic calculation, error-free computing

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DOI: 10.12732/ijpam.v86i1.9 How to cite this paper?
Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2013
Volume: 86
Issue: 1