<p>Solving several polynomial equations in several variables is a hard problem.  Doing so in polynomial time in the average case <a href="http://en.wikipedia.org/wiki/Smale%27s_problems" rel="nofollow">is Smale's 17th problem.</a>  It is unlikely that you will find a fast and simple algorithm for doing so that actually works.</p>

<p>You might look at "Ideals, varieties, and algorithms" by Cox, Little, and O'Shea for an intro to Groebner bases.  Buchberger's algorithm finds a Groebner basis for a given polynomial ideal.  You can find all solutions of a given polynomial system using a Groebner basis for the ideal generated by the polynomials, though the solution comes in a slightly awkward form.</p>

<p>Newton's method is a basic method for solving a system of several nonlinear equations in several variables.  Applied naively, Newton's method is heuristic; it won't always find a solution to a system even if a solution exists.  However, <em>if</em> Newton's method converges, <em>then</em> it converges really fast.  Thus the challenge of the theory problem posed by Smale lies in finding a provably good initial guess to start Newton's method from.</p>

<p><a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.211.3321" rel="nofollow">Beltran and Pardo made considerable progress on Smale's 17th problem</a>, giving an algorithm that works on the average for systems with bounded degree using real-number arithmetic.  This has since been <a href="http://arxiv.org/abs/1205.0869" rel="nofollow">turned into a finite-precision algorithm by Briquel, Cucker, Pena, and Roshchina</a>.  Fascinating as they are, I'm not aware of any implementations, or any attempts at implementations, of these ideas---we're still very, very far away from having <em>usable code</em> for solving systems of polynomial equations.</p>
What is a fast algorithm to find a numeric solution of a system of N polynomial equations on 3 unknown variables?
<p>I'm looking for a fast algorithm to solve a system of N polynomial equations on 3 unknown variables. That is, given 3 functions, <code>F0(x,y,z), F1(x,y,z)... FN(x,y,z)</code>, I want to find <code>x, y, z</code> such that <code>F0(x,y,z) = F1(x,y,z) = ... = FN(x,y,z) = 0</code>. </p>

<p>I've tried finding the solution in several different places, but I could only find very advanced papers on topics such as algebraic geometry or cryptography. What I need, though, is a simple/quick algorithm that returns a fast numerical solution. Is there such algorithm?</p>