Ratio vectors of fourth degree polynomials

Abstract

Let p ( x ) be a polynomial of degree 4 with four distinct real roots r 1 < r 2 < r 3 < r 4 . Let x 1 < x 2 < x 3 be the critical points of p, and define the ratios σ k = x k − r k r k + 1 − r k , k = 1 , 2 , 3 . For notational convenience, let σ 1 = u , σ 2 = v , and σ 3 = 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 in an earlier paper. The main results of this paper involve using the theory of Groebner bases to prove that those conditions are also sufficient.