From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math.num-analysis,sci.math.symbolic
Subject: Re: Solution to a System of Polynomials
Date: 25 Jan 1996 05:40:56 GMT
In article <4e3sc8$hrj@fractal.umd.edu>, Paul T. So wrote:
>I have a set of n nonlinear polynomial equations. In the simplest
>case, the order of the polynomials is two.
>
>a + bx + cx^2 + y=0
>d + ey + fy^2 + z=0
>g + hz + kz^2 + w=0
> .
> .
> .
>
>For a general set of nonlinear equations, I know that Newton-Raphson
>method is good if you have a good initial guess on the solution. Now, if
>the set of nonlinear equations are polynomials (order two or larger),
>does there exist a more GLOBAL and efficient method in obtaining the
>solution. Moreover, in the quadratic case, should I expect a closed
>form solution as in the one dimensional case (the quadratic equation for
>a second order polynomial)?
You need something nicer than "But they're all just quadratic!". Indeed,
any system of polynomials may be treated as a system of quadratic
polynomials, albeit in a greater number of variables. For example, the
Fermat curve for n=3
x^3+y^3=1
is just the projection to the xy-plane of the following curve in 4-space:
xu+yv=1
u-x^2=0
v-y^2=0
(which is given by quadratic equations). So your question just asks,
"Is there a way to determine (in closed form I suppose) all points in
an algebraic set (presumably of dimension zero)?" The only complete
answer would be to use resultants and elimination theory, but that
gets worthlessly hard as the number of solutions grows (which in practice
seems to happen as soon as you move past toy problems.)
It would help if, for instance, a=d=g=... and b=e=h=... and c=f=k=... in
your example; then your system could be viewed as iterations of a quadratic
polynomial. I suppose there might be other special cases of your type for
which some analysis is possible. (By the way, does the last equation
cycle back to ...=x ?)
dave
(posted and mailed as requested.)