We present a complete and practical algorithm which can determine the number of distinct real solutions of a given polynomial system of equations and inequalities with integer coefficients mechanically.Based on this algorithm, a program called nearsolve has been implemented in Maple.The algorithm and program have been successfully applied to many problems with vations backgrounds and to automated discovering and proving for inequality-type theorems.

in this paper,we consider a polynomial differential system of degreen,which was given from a general multimolecular reaction in biochemistry as a theoretical problem of concentration kinetics.The high degree of polynomials involves so many difficulties that we hardly give coordinates of all equilibria,although that is basic for qualitative analysis.Using techniques of decomposition, truncation, and elimination with a computer algebra system,we first give qualitative properties of all equilibria, and then analyze their saddle-node bifurcation and Hopf bifurcation both for real parameters and for integer'parameters.

Making use of the discriminant sequence for polynomials,WR algorithm,Wu's elimination and a partial cylindrical algebraic decomposition,we present here a practical algorithm for automated inequality discovering which can discover new inequalities automatically without requiring to put forward any conjectures beforehand.That is complete for an extensive class of inequality-type theorems.Also this algorithm is applied to the classification of the real physical solutions of geometric constraint problems.Many inequalities with various backgrounds have been discovered or rediscovered by our program,DISCOVERER,which implements the algorithm in Maple.