[GAP Forum] solving equations

[stefan at mcs.st-and.ac.uk]

Dear Forum,

Jan Schneider asked:

> can anybody tell me how to solve multiple equations with GAP?
> I know, it's a pretty easy question, but I just started working with GAP.
> For example, what do I have to do to let GAP calculate 5*x+y = 17, x*y=18?

You can proceed as follows:

First define the variables you need:

gap> x := Indeterminate(Rationals,1);; SetName(x,"x");
gap> y := Indeterminate(Rationals,2);; SetName(y,"y");

Then compute a reduced Groebner basis for the ideal of C[x,y] defined by
your equations, for lex order:

gap> ReducedGroebnerBasis([5*x+y-17,x*y-18],MonomialLexOrdering());
[ y^2-17*y+90, x+1/5*y-17/5 ]

Then solve the first equation y^2-17*y+90 = 0 for y (note that it is
an equation in y, only). Finally, insert the solutions into the second
equation x+1/5*y-17/5 = 0 to compute the possible values of x.

This process works in a similar way general.
-- You can find details in standard textbooks like

Cox, Little, O'Shea: Ideals, Varieties, and Algorithms:
An Introduction to Computational Algebraic Geometry and Commutative Algebra.

If you encounter a univariate polynomial whose Galois group is solvable,
you can use the GAP Package RadiRoot
( see http://www.gap-system.org/Packages/radiroot.html ) by Andreas Distler
to compute representations of its roots in terms of radicals.

There is presently no code in GAP to compute solutions numerically.

Hope this helps,

    Stefan Kohl