Ratio vectors of fourth degree polynomials

Abstract

Let p(x) be a polynomial of degree 4 with four distinct real roots r1<r2<r3<r4. Let x1<x2<x3 be the critical points of p, and define the ratios s_{k}=((x_{k}-r_{k})/(r_{k+1}-r_{k})),k=1,2,3. For notational convenience, let s1=u, s2=v, and s3=w. (u,v,w) is called the ratio vector of p. We prove necessary and sufficient conditions for (u,v,w) to be a ratio vector of a polynomial of degree 4 with all real roots. Most of the necessary conditions were proven by the author in (On the Ratio Vectors of Polynomials, Journal of Mathematical Analysis and Applications 205(1997), 568-576). The main results of this paper involve using the theory of Groebner bases to prove that those conditions are also sufficient.