# how to show the existence of root for a system of polynomial equations?

I come across a system of polynomial equations. Although I can solve it by Mathematica (numerically), I wonder whether it is possible to prove this root indeed exists. Any general method to do it? For example, $f(x_1,x_2)=0$ ; $g(x_1,x_2)=0$ ; $f$ and $g$ are polynomials of $x_1$ and $x_2$, and I compute an approximate root $(x_1^*,x_2^*)$ by some numerical method. How can I show there indeed exists a root around this approximate calculated root? I just wonder whether there exist some general theories in doing so.

• Real root or complex root? How many equations/unknowns? – Igor Rivin Oct 6 '16 at 10:54
• real and two equations with two unknowns – stephenkk Oct 6 '16 at 11:19
• Could you perhaps tell us the system? In some cases you can guess the exact algebraic solution after you solved the system to high accuracy and then check the algebraic solution in exact arithmetic. Sometimes Smale's Alpha Theory helps. – Moritz Firsching Oct 6 '16 at 13:04