Counting positive solutions for polynomial systems with real coefficients

Based on the subresultant polynomial chain and the method presented by Hongguang Fu and Lu Yang et al., we propose a practical algorithm for computing the number of sign changes of the leading coefficients and the constant terms of a generalized Sturm sequence (GSS). With the principal minors sequence of the discrimination matrix of the polynomial and the number of sign changes of the GSS, a formula determining the number of positive solutions of a given bivariate polynomial systems with real coefficients (with a finite number of complex solutions) and its algorithm are presented. Using the techniques of the B-net form of bivariate splines function, this algorithm can be used for computing the number of real intersection points of two piecewise algebraic curves and its distributions in all cells when common points are finite. A piecewise algebraic curve is defined by a bivariate spline function.