Department of Mathematics and Computer Science
Master of Science
We consider the problem of finding, constructing, and classifying nice polynomials. After a short history of previous results, we present a general property of nice polynomials which leads to an important modification of the concept of equivalence classes of nice polynomials. We give several important results on nice symmetric or antisymmetric polynomials with an odd number of roots, which dramatically increase the speed of a computer search for examples. We present complete solutions to the symmetric three root case, the general three root case, and the symmetric four root case. We also give the relations between the roots and critical points for the general four root case and the symmetric five, six, and seven root cases. Using the relations for the general three and four root cases, we state, without proof, the suggested pattern for the relations for the general N root case. We present several important examples antisymmetric polynomials with five distinct roots and the first known examples of nice polynomials with six distinct roots. To conclude our study, we present several open problems and new conjectures suggested by our results, examples, and computer searches.
Groves, Jonathan, "Nice Polynomials" (2004). Masters Theses & Specialist Projects. Paper 526.