# CASA Function: GWalk

Computes the reduced Groebner basis of the polynomial system by means of the Groebner walk algorithm.

### Calling Sequence:

• G := GWalk(F,X1)
• G := GWalk(F,X1,X2)

### Parameters:

F : list(polynom(C, X1))
• A list of polynomials in variables X1 over some coefficient domain C.
X1 : list(name)
• A list of variable names. The list of indeterminates X1 = [x1,x2,...,xn] induces a lexicographical term order where x1>x2>...>xn for which a Groebner basis will be computed.
X2 : list(name)
• A list of the same variable names as occurring in X1 to specify the order of the variables for the intermediate computation of a total degree Groebner basis. The order of the indeterminates in X2 = [x1,x2,...,xn] induces a total degree reverse lexicographical term order (tdeg) with x1>x2>...>xn. If the function is called with only two arguments, X2 defaults to X1.

### Result:

G : list(polynom(D, X1))
• A lexicographical Groebner basis of F. The domain D is the quotient field of the domain C.

### Description:

• The function first computes a Groebner basis with respect to the total degree order (tdeg) induced by the order of variables given in X2 by calling the function grobner[gbasis] and converts it to a lexicographical Groebner basis by means of the Groebner Walk algorithm as described in [26].

### Examples:

> GWalk([t^2+y^2+z^2-1, t^2+z^2-y, t-z], [t,y,z]);

> GWalk([1+x^2+z,x*z+y^2*z],[z,x,y],[x,y,z]);