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Nice symmetric and antisymmetric polynomials

Published online by Cambridge University Press:  01 August 2016

Jonathan Groves
Jonathan Groves*
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027 USA, e-mails: JGroves@ms.uky.edu, Jonny77889@yahoo.com
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We first present a general property of nice polynomials, which leads to an important modification of the concept of equivalence classes of nice polynomials. We then establish several properties of nice symmetric and antisymmetric polynomials with an odd number of distinct roots. We give a complete description of all nice symmetric and antisymmetric polynomials with three distinct roots. We then apply our results to nice symmetric sextics and octics. As an important application, all the properties we have established have dramatically increased the speed of a computer search for examples, allowing us to discover the first examples of nice symmetric and antisymmetric polynomials with five distinct roots and the first two examples of nice polynomials with six distinct roots.

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Articles
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The Mathematical Gazette , Volume 92 , Issue 525 , November 2008, pp. 437 - 453
Copyright
Copyright © The Mathematical Association 2008

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References

1. Chappie, M. A cubic equation with rational roots such that it and its derived equation also has rational roots, Bull. Math. Teachers Secondary Schools, 11 (1960) pp. 5–7. (Republished in Aust. Senior Math. J. 4(1) (1990) pp. 57–60.)Google Scholar
2. Nowakowski, Richard Unsolved problems, 1969–1999, Amer. Math. Monthly 106 (10) (1999) pp. 959–962.Google Scholar
3. Buchholz, Ralph H. and MacDougall, James A. When Newton met Diophantus: A study of rational-derived polynomials and their extensions to quadratic fields, J. Number Theory 81: (2000) pp. 210–233.Google Scholar
4. Evard, Jean-Claude Polynomials whose roots and critical points are integers, Submitted and posted on the Website of Arxiv Organization at the address http://arxiv.org/abs/math/0407256.Google Scholar
5. Groves, Jonathan Nice Polynomials, Master’s thesis written at Western Kentucky University, July 2004.Google Scholar
6. Groves, Jonathan Nice polynomials with three roots, Math. Gaz. 92 (March 2008) pp. 1–7.Google Scholar
7. Groves, Jonathan Nice polynomials with four roots, Far East J. of Math. Sci. 27(1), (2007) pp. 29–42.Google Scholar
8. Groves, Jonathan A new tool for the study of D-nice polynomials, Version of 1 August 2007, http://www.math.UKY.edu/∼jgroves.Google Scholar
9. Caldwell, Chris K. Nice polynomials of degree 4, Math. Spectrum 23(2) (1990) pp. 36–39.Google Scholar