On the Exact Number of Zeros of Certain Even Degree Polynomails : A Sharkovsky Theorem Approach

Abstract

The problem of finding or characterizing the zeros of polynomial functions has a long history. The obtained results varied from theory, algorithms and numerical simulations. Historically, this study began with the fundamental theorem of algebra proved by Gauss. The most known results in this direction is the fact that a real polynomial of degree n has at most n real zeros. There is also the so called Descartes rule of signs concerning the number of positive zeros. Some generalizations of Descartes rule are know such as the BudanFourier theorem that gives an upper bound for the number of zeros of a polynomial. Also, the Sturm’s theorem that gives a method for determining the exact number of zeros in an interval [Hen, chapter 6] and [Hou, chapter 2]. Recent results uses the classical Enestrom-Kakeya theorem to restricts the location of the zeros based on a condition imposed on the coefficients of the polynomial under invistigation. See [Bre] and references therain. In this paper, we will use a dynamical system result concerning periodic points of a continuous function. The result is called Sharkovsky theorem [Sha1, Sha2, Sha3, Sha4, Sha5] that gives a complete description of possible sets of periods for continuous mappings defined on an interval. The interval need not be closed or bounded. The main idea used here is the notion of