gbasis takes at least two arguments
Optionnal arguments may be used to specify the ordering and
algorithms. By default, the ordering is lexicographic (with respect to the
list of variable names ordering)
and the polynomials are written in decreasing power orders with
respect to this order.
For example, the output will be like
...+x^{2} y^{4} z^{3}+x^{2} y^{3} z^{4}+... if the second argument is [x,y,z] because
(2,4,3)>(2,3,4) but the output would be like
...+x^{2} y^{3}z^{4}+x^{2} y^{4} z^{3}+... if the second argument is [x,z,y].
gbasis returns a Gröbner basis of the polynomial ideal spanned
by these polynomials.
Property
If I is an ideal and if (G_{k})_{k ∈ K} is a Gröbner basis of this
ideal I then, if F is a non zero polynomial in I, the greatest monomial
of F is divisible by the greatest monomial of one of the G_{k}.
In other words, if you do an euclidian division of F≠ 0
by the corresponding G_{k}, take the remainder of this division, do
again the same and so on, at some point you get a null remainder.
Input :
^
2,x^
2-2*x*y],[x,y])
Output :
^
2+-4*y^
2,2*x*y-y^
2,-(3*y^
3)]
As indicated above, gbasis may have more than 2 arguments :
Input :
Output
[x3^3-1,-x2^2-x2*x3-x3^2,x1+x2+x3]