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

Year 2007, Volume: 2 Issue: 2, 208 - 225, 01.12.2007

Jonathan Groves

Abstract

Let D be any integral domain of any characteristic. A polynomial p(x) ∈ D[x] is D-nice if p(x) and its derivative p′(x) split in D[x]. We give a complete description of all D-nice symmetric polynomials with four roots over integral domains D of any characteristic not equal to 2 by giving an explicit formula for constructing these polynomials and by counting equivalence classes of such D-nice polynomials. To illustrate our results, we give several examples we have found using our formula. We conclude by stating the open problem of finding all D-nice symmetric polynomials with four roots over integral domains D of characteristic 2 and all D-nice polynomials with four roots over all integral domains D of any characteristic.

Keywords

Critical point, D-nice polynomial, integral domain, nice polynomial, polynomial, root, symmetric polynomial

References

• 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 Şelds, J. Number Theory, 81 (2000), 210-233.
• C.K. Caldwell, Nice polynomials of degree 4, Math. Spectrum, 23(2) (1990), 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.NT/0407256.
• J. Groves, Nice symmetric and antisymmetric polynomials, To appear in Math. Gazette. J. Groves, Nice polynomials with three roots, To appear in Math. Gazette. J. Groves, Nice polynomials with four roots, To appear in Far East J. Math. Sci. J. Groves, A new tool for the study of D-nice polynomials, Version of January , 2007.
• 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.
• Karl Zuser, Uber eine gewisse Klasse von ganzen rationalen Funktionen 3. Grades, Elem. Math, 18 (1963), 101-104. Jonathan Groves
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