A set of polynomials {F1,…,FN} generate an ideal I; namely, I is the set of all linear combinations of the Fj. Given such an ideal, a Gröbner basis for I is a subset G={G1,…,Gn} of I such that for any F in I, there is a Gk in G such that the leading monomial of Gk divides the leading monomial of F. (Note that the leading monomial depends on a fixed ordering of the monomials.)
If G is a Gröbner basis for such an ideal I, then for any nonzero F in I, if you do a Euclidean division of F by the corresponding Gk, take the remainder of this division, do again the same and so on, at some point you get a remainder of zero.
Let I be the ideal generated by {x3−2xy, x2y−2y2+x} with the standard lexicographic order on the monomials. One Gröbner basis for I is
G={g1(x,y)=x2,g2(x,y)=x y,g3(x,y)=2 y2−x}. |
Consider the element F(x,y)=2x2y−3x2+6xy−4y2+2x of I. The leading monomial x2y of F(x,y) is divisible by the leading monomial x2 of g1(x,y). Dividing F(x,y) by g1(x,y) leaves a remainder of R1(x,y)=6xy−4y2+2x. The leading monomial of R1(x,y), which is xy, is divisible by the leading monomial of g2(x,y), which is xy. Dividing R1(x,y) by g2(x,y) leaves a remainder of R2(x,y)=−4y2+2x. Finally, the leading monomial of R2(x,y), which is y2, is divisible by the leading monomial of g3(x,y), which is y2. Dividing R2(x,y) by g3(x,y) leaves a remainder of 0.
The gbasis command computes Gröbner bases.
A value of true is recommended, but requires that CoCoA support be compiled into Xcas.
Note that the lexicographic order depends on the order the variables are given in vars. For example, if vars=[x,y,z], then x^2*y^4*z^3 comes before x^2*y^3*z^4, but if vars=[x,z,y], then x^2*y^4*z^3 comes after x^2*y^3*z^4.
gbasis([2*x*y-y^2,x^2-2*x*y],[x,y]) |
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gbasis([x1+x2+x3,x1*x2+x1*x3+x2*x3,x1*x2*x3-1],[x1,x2,x3],tdeg,with_cocoa=false) |
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