Research On Relations Between Inequalities Of Polynomials And Eigenvalue Optimization Problems

In this paper we mainly study the relation between the coefficients of the prod-uct of any two nonzero polynomials and the ones of these polynomials. This paper consists of five chapters:In Chapter1, we summarize the research on the problems of the polynomial optimization, the polynomial fractional optimization and the estimate of polynomial coefficients of at home and abroad.In Chapter2, first, the optimization problems studied in [33] are transformed into the polynomial fractional optimization which is turned into the polynomial optimizations, and an example is solved by GloptiPoly. Secondly, because it was proved that the optimization problems studied in [33] is equivalent to the eigenvalue optimization of the autocorrelation matrix, an algorithm for the problems is pre-sented here according to the character of the autocorrelation matrix. An example is computed by the software programmed by matlab according to the algorithm.In Chapter3, the part of the results in [33] is generalized to the case of poly-nomials in more than1variables.In Chapter4, the problems on the polynomials in n variables is intuitively showed by the ones on the polynomials in2variables.In Chapter5, the problems studied in this paper are summarized, and the future research problems are proposed.