Gröbner basis

### 11.5.1 Gröbner basis

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.

#### Example

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.

- gbasis takes two mandatory arguments and three optional arguments:
  - polys, a list of polynomials.
  - vars, a list of the variable names.
  - Optionally, order, which can be one of:
    - plex, to order the monomials lexicographically (this is the default).
    - tdeg, to order the monomials first by total degree then by lexicographic order.
    - revlex, to order the monomials reverse lexicographically.
  - Optionally, with_cocoa=boolean, where boolean can be true or false. A value of true means to use the CoCoA library to compute the Gröbner basis, a value of false means not to use it.

    A value of true is recommended, but requires that CoCoA support be compiled into Xcas.

  - Optionally, with_f5=boolean, where boolean can be true or false. A value of true means to use the F5 algorithm of the CoCoA library, a value of false means not to use it. If this is true, then the polynomials are homogenized and so the specified order is not used.
- gbasis(polys,vars ⟨,order,with_cocoa=boolean, with_f5=boolean ⟩) returns a Gröbner basis of the ideal spanned by polynomials in polys.

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.

#### Examples

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

| ⎡ ⎣ | y3,x2−y2,2 x y−y2 | ⎤ ⎦

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

| ⎡ ⎣ | x33−1,−x22−x2 x3−x32,x1+x2+x3 | ⎤ ⎦