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