From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: Can Maple return only real roots
Date: 5 Nov 1999 05:50:11 GMT
Newsgroups: sci.math.symbolic
Keywords: example of elimination theory showing great swell of Groebner bases
In article <1107_941722743@k313a.krannert>,
Priyo Chatterjee wrote:
>I'm trying to extract the real roots of the following simultaneous
>equation system. Maple does solve it and gives two nice (real) roots
>for x and y and some other roots in "Root_Of" notation. What I need to
>know is whether the other roots are real or complex. I tried
>specifying (using the assume command) that both x and y are real.
>One can assume that h>0, L>0, alpha>0 and c>0.
>
>eqn1:=(alpha+y*h-L*(y+1))^2*(y-x+1)-c*(x+y+1)^3;
>
>eqn2:=(alpha+x*L-h*(x+1))^2*(x-y+1)-c*(x+y+1)^3;
Maybe it's just me but I think it better to reserve the word "roots" for
solutions of a single univariate polynomial. Several equations in several
unknowns determine varieties --- algebraic subsets of affine space. In this
case you expect the solution set to be a finite set of points (x,y). I take
it you want to know how many of those points have both coordinates
real.
Well, you can use elimination (e.g. Maple's "eliminate" command) to reduce this
to a one-variable problem: the set of y-coordinates of your points
satisfy a certain sextic, which has a linear factor and a quintic factor
(which generically has Galois group Sym(5), i.e., you can't expect any kind
of further simplification in general). From the linear factor we learn that
there is one easily described point in your solution set: (x,y)=
( (alpha - h)/(h - L) , (L - alpha)/(h - L) ). The other five points have
real coordinates iff the y-coordinate is real; obviously this must be true
for at least one root of the quintic, and can happen for three of the roots.
I suppose it can be true that all five of the roots are real, so that there
is a grand total of 6 real points in your original variety, but I didn't
happen to find one with simple experimentation.
Of course, some combinations of the initial parameters can lead to
degenerate situations I didn't touch on here.
Special bonus for the cognoscenti: from the ideal (eqn1, eqn2) in
Q[ x, y, alpha, h, L, c ] may be computed a Groebner basis (with the
lexicographic ordering on the variables). Magma did this almost instantly,
BUT -- the answer has dozens of generators, takes 15000 lines to print,
and occupies a megabyte of disk space in text form!
dave