From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: Polynomial systems of equations Date: 31 Mar 1997 22:05:25 GMT In article <33385039.4371@mail.mia.bellsouth.net>, harden wrote: >What are the theorems concerning polynomial systems of equations that >correspond to the fundamental theorem of algebra for single equations? >How many solutions are there? Do solutions always exist? etc. You want Bezout's theorem, and Algebraic Geometry more generally. Roughly speaking if you have two equations in two variables the solution set is finite (the number of points is typically the product of the degrees of the polynomials). It's very difficult to set up carefully, yet the geometry introduced along the way is beautiful. >Also, what are the most efficient algorithms for numerically >approximating the solutions to polynomial systems of equations? >Can some adaptation of Newton's method be used here? Sure. Think again of the example F: R^2 -> R^2 (or F: C^2 -> C^2 if you want complex solutions). If P0 = (x0,y0) is a point near a solution, you can in general get a point P1 nearer to the solution by computing P1 = P0 - (F'(P0))^(-1) * F(P0). Here F' is the 2x2 Jacobian matrix, which you evaluate at P0, invert, and multiply by the vector F(P0). All the caveats associated with Newton's method in R^1 apply here as well, of course. dave