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

Authors : Jonathan GROVES

Pages : 78-105

View : 13 | Download : 10

Publication Date : 2009-06-01

Article Type : Research Paper

Abstract :Let D be an integral domain of any characteristic. We say that pinsert ignore into journalissuearticles values(x); ∈ D[x] is D-nice if pinsert ignore into journalissuearticles values(x); and its derivative p0insert ignore into journalissuearticles values(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 pinsert ignore into journalissuearticles values(x); with four distinct roots. We consider the following three cases separately: insert ignore into journalissuearticles values(1); D has characteristic 0, insert ignore into journalissuearticles values(2); D has characteristic p > 0 and the degree of pinsert ignore into journalissuearticles values(x); is not a multiple of p, and insert ignore into journalissuearticles values(3); D has characteristic p > 0 and the degree of pinsert ignore into journalissuearticles values(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 pinsert ignore into journalissuearticles values(x);. To conclude, we state several problems about D-nice polynomials with four roots that remain unsolved.
Keywords : D nice polynomial, nice polynomial