I have a system of polynomials, where the first one is a multivariate linear polynomial, but the rest are univariate quadratic polynomials. How would I solve such a system (finding one or all solutions, or showing there are no solutions)? For example,
$$17x+16y-5z-67=0 \\ x^2+3x-5=0 \\ 4y^2-7y-4=0 \\ z^2-6z-3=0$$
Best Answer
The system has no solution. This can be seen by computing a Groebner basis, for example. But also a direct approach is possible. We can compute $x,y,z$ from the second, third and last equation (two solutions each) and then substitute it into the first one. Even if we would replace the first equation by $$ 17x+16y-5z-a=0 $$ where $a$ is an integer, there is no solution.