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 ([3]). The main results of this paper involve using the theory of Groebner bases to prove that those conditions are also sufficient.