D-NICE POLYNOMIALS WITH FOUR ROOTS OVER INTEGRAL DOMAINS D OF ANY CHARACTERISTIC

Abstract

Let D be an integral domain of any characteristic. We say that p(x) ∈ D[x] is D-nice if p(x) and its derivative p0(x) split in D[x]. We begin by presenting a new equivalence relation for D-nice polynomials over integral domains D of characteristic p > 0, which leads to an important modification of our definition of equivalence classes of D-nice polynomials. We then present a partial solution to the unsolved problem of constructing and counting equivalence classes of D-nice polynomials p(x) with four distinct roots. We consider the following three cases separately: (1) D has characteristic 0, (2) D has characteristic p > 0 and the degree of p(x) is not a multiple of p, and (3) D has characteristic p > 0 and the degree of p(x) is a multiple of p. In all these cases we give formulas for constructing some examples. In the final case we also count equivalence classes of D-nice polynomials for certain choices of the multiplicities of the roots of p(x). To conclude, we state several problems about D-nice polynomials with four roots that remain unsolved.

Keywords

• D-nice polynomial, nice polynomial

References

1. T. Bruggeman and T. Gush, Nice cubic polynomials for curve sketching, Math. Magazine, 53(4) (1980), 233-234.
2. R.H. Buchholz and J.A. MacDougall, When Newton met Diophantus: A study of rational-derived polynomials and their extensions to quadratic fields, J. Num- ber Theory, 81 (2000), 210-233.
3. C.K. Caldwell, Nice polynomials of degree 4, Math. Spectrum, 23(1990), 36-39.
4. M. Chapple, A cubic equation with rational roots such that it and its derived equation also has rational roots, Bull. Math. Teachers Secondary Schools, 11 (1960), 5-7 (Republished in Aust. Senior Math. J. 4(1) (1990), 57-60).
5. J.C. Evard, Polynomials whose roots and critical points are integers, Sub- mitted and posted on the Website of Arxiv Organization at the address http://arxiv.org/abs/math/0407256.
6. J. Groves, Nice symmetric and antisymmetric polynomials, Math. Gazette, (525) (2008), 437-453.
7. J. Groves, Nice polynomials with three roots, Math. Gazette, 92(523)(2008),1-7.
8. J. Groves, Nice polynomials with four roots, Far East J. Math. Sci., 27(1) (2007), 29-42.

9. J. Groves, A new tool for the study of D-nice polynomials, Version of August , 2007, Posted on the author’s website http://www.math.uky.edu/˜jgroves.
10. J. Groves, D-nice symmetric polynomials with four roots over integral domains D of any characteristic, Inter. Electron. J. Algebra, 2 (2007), 208-225.
11. R.K. Guy, Unsolved problems come of age, Amer. Math. Monthly, 96(10) (1989), 903-909.
12. R. Nowakowski, Unsolved problems, 1969-1999, Amer. Math. Monthly, 106(10) (1999), 959-962. Jonathan Groves
13. University of Kentucky Department of Mathematics Patterson Office Tower 713 Lexington, KY 40506-0027 e-mails: [e-mail], [e-mail]