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

Authors : Jonathan Groves

Pages : 208-225

View : 13 | Download : 13

Publication Date : 2007-12-01

Article Type : Research Paper

Abstract :Let D be any integral domain of any characteristic. A polynomial pinsert ignore into journalissuearticles values(x); ∈ D[x] is D-nice if pinsert ignore into journalissuearticles values(x); and its derivative p′insert ignore into journalissuearticles values(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