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

Year 2009, Volume: 5 Issue: 5, 78 - 105, 01.06.2009

Jonathan Groves

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

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