D-NICE POLYNOMIALS WITH FOUR ROOTS OVER INTEGRAL DOMAINS D OF ANY CHARACTERISTIC
D-NICE POLYNOMIALS WITH FOUR ROOTS OVER INTEGRAL DOMAINS D OF ANY CHARACTERISTIC
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,
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- T. Bruggeman and T. Gush, Nice cubic polynomials for curve sketching, Math. Magazine, 53(4) (1980), 233-234.
- 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.
- C.K. Caldwell, Nice polynomials of degree 4, Math. Spectrum, 23(1990), 36-39.
- 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).
- 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.
- J. Groves, Nice symmetric and antisymmetric polynomials, Math. Gazette, (525) (2008), 437-453.
- J. Groves, Nice polynomials with three roots, Math. Gazette, 92(523)(2008),1-7.
- J. Groves, Nice polynomials with four roots, Far East J. Math. Sci., 27(1) (2007), 29-42.
- 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.
- J. Groves, D-nice symmetric polynomials with four roots over integral domains D of any characteristic, Inter. Electron. J. Algebra, 2 (2007), 208-225.
- R.K. Guy, Unsolved problems come of age, Amer. Math. Monthly, 96(10) (1989), 903-909.
- R. Nowakowski, Unsolved problems, 1969-1999, Amer. Math. Monthly, 106(10) (1999), 959-962. Jonathan Groves
- University of Kentucky Department of Mathematics Patterson Office Tower 713 Lexington, KY 40506-0027 e-mails: JGroves@ms.uky.edu, Jonny77889@yahoo.com
- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI
Sayıdaki Diğer Makaleler
D-NICE POLYNOMIALS WITH FOUR ROOTS OVER INTEGRAL DOMAINS D OF ANY CHARACTERISTIC
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