TITLE: how to show the existence of root for a system of polynomial equations?
QUESTION [6 upvotes]: 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.

REPLY [8 votes]: The magic words are: Cylindrical Algebraic Decomposition.