How to find roots of a system of multivariate polynomials?

Question

I am trying to find the roots of a system of 3 multivariate polynomials with 3 variables. The polynomials are really 'ugly'. So far I have tried to find a Groebner Basis in Maple and got a Groebner Basis with 29 elements (the length of output exceeds the limit). I want to find roots in the interval [0,1] with x<y<z. Is there a way to find solutions? Some of the polynomials in the Basis are of order 11 and I can't find a single variable polynomial in the basis. Is there an efficient way to find such roots? Or should I do someting completely different?

Best,

fabs

Answer 1

If all you want is some isolated candidate solutions (not whole algebraic or analytic surfaces) then a numerical optimization may be good enough.

Here is a numerical approach you can try

1. Start at a random point.
2. Select 1 variable to let vary. Let us say it is variable nr $k$. Fix the others. This gives you three one-variable polynomials in variable $k$.
3. Seek to minimize sum of squares. This gives you one polynomial.
4. Select a numerical root finder for your square sum polynomial. Perhaps Newton-Rhapson? Let it run until it comes under some threshold or stagnates.
5. When stagnates or under threshold but too far from 0, jump back to 2 and switch variable to let vary but keep the point you are at as start for the new search.

If you ever find square sum = 0 then you have found a root.

However as you have same number of equations as variables it is quite possible that there are no solutions.

For example : Just consider three sphere's surfaces which don't intersect even pair-wisely.

If the equations of the sun, the earth and the moon surfaces intersected in just any one point we would be rather screwed. Bad enough if only two of them did.

Answer 2

An alternative to Groebner bases when considering systems over the reals that have constraints is to use either 'regular chains' or 'cylindrical algebraic decompositions', both of which Maple implements - see the RegularChains package (the help page will direct you as to good starting commands). If it turns out your system only have isolated zeroes, then the sub-package SemiAlgebraicSetTools should be extra useful.