In this puzzle, we asked: suppose and . What then is ?

First let us solve this by hand. Since , . So

Hence it follows that

But since , this means that , so if we must have or vice versa. In any case, it follows that .

The reason I wanted to present this problem is that it shows the power and beauty of Gröbner bases. Here is how to solve this problem in MAPLE. Define . Then the question is: what equation does satisfy? So we set up the three equations, two of which are the data of the problem and the last one is the definition of . Crank up MAPLE…

with(Groebner):

e1 := a+b-1:

e2 := a^3+b^3-1:

e3 := a^17+b^17-z:

Basis([e1,e2,e3],plex(z,a,b));

[-b + b , a + b – 1, z – 1]

The Basis([e1,e2,e3],plex(z,a,b)) command computes the Gröbner basis of (the ideal defined by) this set of equations (with the right lexicographic ordering) and *has* to give a univariate polynomial in . The resulting three equations are equivalent (in terms of their zeroes) to the original ones. The only zero of is .

I leave many things unexplained on purpose: if you are interested, the help system of MAPLE and all the resources of the web are at your disposal.

(MG)

### Like this:

Like Loading...

*Related*