Company
Mathematica Solutions to the ISSAC '97 Systems Challenge

Wolfram Research, Inc.


Problem 6

Find a lexicographic Gröbner basis for the following polynomial system.

[Graphics:ISSACChallengegr104.gif]

Result
[Graphics:ISSACChallengegr105.gif]


Method: Use a built-in function.

The default term ordering for GroebnerBasis is the lexicographic ordering, so the input is straightforward.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr106.gif]

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr107.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr108.gif]

The basis consists of a univariate polynomial in [Graphics:ISSACChallengegr109.gif] and three linear polynomials in [Graphics:ISSACChallengegr110.gif], [Graphics:ISSACChallengegr111.gif], and [Graphics:ISSACChallengegr112.gif].

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr113.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr114.gif]

A numerical version of the Gröbner basis can be calculated more than 150 times faster.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr115.gif]

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr116.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr117.gif]

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr118.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr119.gif]

This shows that the numerically calculated Gröbner basis and the exact one are the same.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr120.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr121.gif]