Real Solution Bound For Fewnomial Polynomial System With The Application

Abstract we research a new bound of real solutions to a fewnomial system consisting of n polynomials in n variables having a total of n+k+1 distinct monomial, and on the basis of this, we also study the bound of the contact components of hypersurface’s num-ber, the connected components of hypersurface’s number, the arcs of implicit curves’ number and the Betti number. By way of decreasing constant term and constructing some new polynomial functions F1*,..., Fk*, we can obtain the bound. We can reduce the constant term by using function theory and mathematical induction; According to Khovanskii-Rolle theorem and Gale dual relationship, we construct some new polyno-mials F1*,...,Fk* and also obtain the bound of the number of real solutions to the system F1*=...= Fk*= 0 by Bezout theorem. With the increasing of n and k, specially, the improvement of the new bound of real solutions is more obvious in comparison with Bate’s bound of the number of solutions when k >> n.Firstly, we introduce the relevant background and research progress. Secondly, the paper describes the background knowledge and prove the feasibility of the positive real solution number bound,and also contrast with Bihan’s bound by analyzing the exam-ple. Thirdly, based on the positive real solution number bound, we gain the bound of the number of real solutions to the system, and then prove the feasibility, and also contrast with Bate’s bound by combining charts and picture; Fourthly, based on the bound of the numbers of positive real solution and real solution, we gain the bound of the number of contact components of hypersurface, the numbers of connected components of hy-persurface, the numbers of arcs of implicit curves and the sum of Betti numbers. Lastly, we analyses the advantages of bound and the disadvantage, and the direction that we can study latter.