Details:
Title  Ratio vectors of fourth degree polynomials  Author(s)  Alan Horwitz  Type  Article in Journal  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.  Keywords  Polynomial, Real roots, Groebner basis  ISSN  0022247X 
URL 
http://www.sciencedirect.com/science/article/pii/S0022247X05005457 
Language  English  Journal  Journal of Mathematical Analysis and Applications  Volume  313  Number  1  Pages  132  141  Year  2006  Edition  0  Translation 
No  Refereed 
No 
