On the Number of Solutions to Polynomial Systems of Equations
C. B. Garcia, T. Y. Li · SIAM Journal on Numerical Analysis · 1980
It is shown that for almost every system of n polynomial equations in n complex variables, the number of solutions is equal to $q \equiv \Pi _{i = 1}^n q_i $, where $q_i $ is the degree of equation i. The proof of this result is done in such a way that all q solutions can be explicitly calculated for almost all such systems. It is further shown that if the polynomial system obtained by retaining only the terms of degree $q_i $ in each equation i has only the trivial solution, then the number of solutions is equal to q.