I heard that something called a Groebner Basis can be used to find the solutions of a system of polynomial equations like $p_1(x_1,\ldots,x_m)=0,\ldots,p_n(x_1,\ldots,x_m)=0$
$$p_1(x_1,\ldots,x_m)=0,\ldots,p_n(x_1,\ldots,x_m)=0$$ $$q_1(x_1,\ldots,x_m)>0,\ldots,q_r(x_1,\ldots,x_m)>0$$