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2.27.1  Gröbner basis : gbasis

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 ...+x2 y4 z3+x2 y3 z4+... if the second argument is [x,y,z] because (2,4,3)>(2,3,4) but the output would be like ...+x2 y3z4+x2 y4 z3+... 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 (Gk)kK 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 Gk. In other words, if you do an euclidian division of F≠ 0 by the corresponding Gk, take the remainder of this division, do again the same and so on, at some point you get a null remainder.

Input :

gbasis([2*x*y-y^2,x^2-2*x*y],[x,y])

Output :

[4*x^2+-4*y^2,2*x*y-y^2,-(3*y^3)]

As indicated above, gbasis may have more than 2 arguments :

Input :

gbasis([x1+x2+x3,x1*x2+x1*x3+x2*x3,x1*x2*x3-1], [x1,x2,x3],tdeg,with_cocoa=false)

Output

[x3^3-1,-x2^2-x2*x3-x3^2,x1+x2+x3]

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