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Ratio vectors of fourth degree polynomials

Alan Horwitz
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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.

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Alan Horwitz. 2003-12-22. Ratio vectors of fourth degree polynomials. https://arxiv.org/abs/math/0312404

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