Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T21:09:37.912Z Has data issue: false hasContentIssue false
  • English
  • Français

STRICTLY REAL FUNDAMENTAL THEOREM OF ALGEBRA USING POLYNOMIAL INTERLACING

Part of: Real and complex fields Polynomials, rational functions

Published online by Cambridge University Press:  18 January 2021

SOHAM BASU Open the ORCID record for SOHAM BASU [Opens in a new window]
SOHAM BASU*
Affiliation:
School of Engineering, École Polytechnique Fédérale de Lausanne, Lausanne1015, Switzerland
*
e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Without resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses interlacing of bivariate polynomials similar to Gauss’s first proof of the fundamental theorem of algebra using complex numbers, but in a different context of division residues of strictly real polynomials. This shows the sufficiency of basic real analysis as the minimal platform to prove the fundamental theorem of algebra.

Keywords

fundamental theorem of algebrapolynomial interlacing

MSC classification

Primary: 12D05: Polynomials
Secondary: 12D10: Polynomials 26C10: Polynomials
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 2 , October 2021 , pp. 249 - 255
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Australian Mathematical Publishing Association Inc. 2021.

References

Aliabadi, M., ‘A note on the fundamental theorem of algebra’, Bull. Aust. Math. Soc. 97(3) (2018), 382–385.CrossRefGoogle Scholar
Basu, S. and Velleman, D. J., ‘On Gauss’s first proof of the fundamental theorem of algebra’, Amer. Math. Monthly 124(8) (2017), 688–694.CrossRefGoogle Scholar
Gauss, C. F., Demonstratio Nova Theorematis Omnem Functionem Algebraicam Rationalem Integram Unius Variabilis In Factores Reales Primi Vel Secundi Gradus Resolvi Posse, PhD Thesis, Helmstedt, 1799.Google Scholar
Perrucci, D. and Roy, M. F., ‘Quantitative fundamental theorem of algebra’, Q. J. Math. 70(3) (2019), 1009–1037.CrossRefGoogle Scholar
Pukhlikov, A. V., “Real” Proof of the Fundamental Theorem of Algebra, Matematicheskoe Prosveshchenie Series 3, 1 (Moscow Center for Continuous Mathematical Education, Moscow, 1997), 85–89 (in Russian).Google Scholar
Pushkar, P. E., On Some Topological Proofs of the Fundamental Theorem of Algebra’, Matematicheskoe Prosveshchenie Series 3, 1 (Moscow Center for Continuous Mathematical Education, Moscow, 1997), 90–95 (in Russian).Google Scholar
Rudin, W., Principles of Mathematical Analysis, 3rd edn (McGraw-Hill, New York, 1976), Sect. 1.10.Google Scholar
Shipman, J., ‘Improving the fundamental theorem of algebra’, Math. Intelligencer 29 (2017), 9–14.CrossRefGoogle Scholar
Sturm, J. C. F., ‘Mémoire sur la résolution des équations numériques’, Bull. Sci. Férussac 11 (271) (1829), 419–425.Google Scholar