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** monter:** Gröbner basis and Gröbner
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##

Gröbner reduction : `greduce`

`greduce` has three arguments : a multivariate
polynomial,
a vector made of polynomials which is supposed to be a Gröbner
basis, and a vector of variable names.

`greduce` returns the reduction of
the polynomial given as first argument
with respect to the Gröbner basis given as the second argument.
It is 0 if and only if the polynomial belongs to the ideal.
Input :

`greduce(x*y-1,[x``^`

2-y`^`

2,2*x*y-y`^`

2,y`^`

3],[x,y])

Output :
`y``^`

2-2

that is to say
*xy* - 1 = (*y*^{2} -2) mod *I* where *I* is the ideal
generated by the Gröbner basis
[*x*^{2} - *y*^{2}, 2*xy* - *y*^{2}, *y*^{3}], because
*y*^{2} - 2 is the euclidian division rmainder of 2(*xy* - 1) by
*G*_{2} = 2*xy* - *y*^{2}.

Like `gbasis` (cf. 1.27.1),
`greduce` may have more than 3 arguments to specify ordering and
algorithm if they differ from the default (lexicographic ordering).
Input :

`greduce(x1``^`

2*x3`^`

2,[x3`^`

3-1,-x2`^`

2-x2*x3-x3`^`

2,x1+x2+x3], [x1,x2,x3],tdeg)

Output
`x2`

giac documentation written by Renée De Graeve and Bernard Parisse