# CASA Function: mgbasis

Compute the normed reduced Groebner basis.

### Calling Sequence:

• GB := mgbasis(F, X)
• GB := mgbasis(F, X, torder1)
• GB := mgbasis(F, X, torder1, torder2)

### Parameters:

F : list(list(polynom(rational)))
• A list of polynomial tuples.
X : list(name)
• A list of indeterminates.
torder1 : name
• A power-product-tuple ordering. Either term (for term first) or index (for index first - default).
torder2 : name
• A power-product ordering. Either plex (for pure lexicographic) or tdeg (for total degree - default).

### Result:

GB : list(list(polynom(rational)))
• The normed reduced Groebner basis of F.

### Description:

• The command mgbasis(F, X, torder1, torder2) computes the normed reduced Groebner basis of F with respect to the indeterminates X and the given orderings. The algorithm used in mgbasis is described as Alg.13 in [2].
• The polynomial tuples in F are represented as lists of polynomials.
• If X has the form [x1, x2, ..., xn], then using the pure lexicographic ordering this is interpreted as x1 > x2 > ... > xn. Within the total degree ordering, ties are broken by inverse lexicographic order.

### Examples:

> F := [[x*y-1,x+2], [y^2+x+1,y-1]]:

> mgbasis(F, [x,y]);

> mgbasis(F, [y,x], term, plex);