Ratio vectors of fourth degree polynomials

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 σk = xk−rk rk+1−rk , k = 1, 2, 3. For notational convenience, let σ1 = u, σ2 = v, and σ3 = 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.  2005 Elsevier Inc. All rights reserved