Computation of a reduced Gröbner basis
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groebner::gbasis(polys
, <order
>, options
)
groebner::gbasis(polys)
computes a reduced
Gröbner basis of the ideal generated by the polynomials in the
list polys
.
The rules laid down in the introduction to the groebner package concerning the polynomial types and the ordering apply.
The polynomials in the list polys
must all
be of the same type. In particular, do not mix polynomials created
via poly
and
polynomial expressions!
The ordering strategy indicated by Reorder
is
used automatically when polynomial expressions are used.
We demonstrate the effect of various input formats. First, we use polynomial expressions to define the polynomial ideal. The Gröbner basis is returned as a list of polynomial expressions:
groebner::gbasis([x^2  y^2, x^2 + y], LexOrder)
Next, the same polynomials are defined via poly
. Note that poly
fixes the ordering of the variables.
groebner::gbasis([poly(x^2  y^2, [x, y]), poly(x^2 + y, [x, y])], LexOrder)
Changing the ordering of the variables in poly
changes the lexicographical ordering.
This results in a different basis:
groebner::gbasis([poly(x^2  y^2, [y, x]), poly(x^2 + y, [y, x])], LexOrder)
With Reorder
the ordering of the variables
may be changed internally:
groebner::gbasis([poly(x^2  y^2, [x, y]), poly(x^2 + y, [x, y])], LexOrder, Reorder)
Polynomials over arbitrary fields are allowed. In particular, you can use the field of rational functions in some given variable(s):
F := Dom::Fraction(Dom::DistributedPolynomial([y])): F::Name := "Q(y)": groebner::gbasis( [poly(y*z^2 + 1, [x, z], F), poly((y^2 + 1)*x^2  y  z^3, [x, z], F)])
delete F:

A list or set of polynomials or polynomial expressions of the
same type. The coefficients in these polynomials and polynomial expressions
can be arbitrary arithmetical expressions. If 

One of the identifiers 

With this option, 

With this option, 

With this option With this option the variables are sorted internally such that they have a "heuristic optimal" ordering. Consequently, the ordering of the variables in the output polynomials may differ from their ordering in the input polynomials. For details on the ordering strategy, see W. Boege, R. Gebauer und H. Kredel: "Some Examples for Solving Systems of Algebraic Equations by Calculating Groebner Bases" im J. Symbolic Comp. (1986) Vol. 1, 8398. Reordering is always applied when polynomial expressions are used for input. 

Option, specified as This option sets the normalizing routine to The method By default, 

Option, specified as This option is equivalent to passing 
List of polynomials. The output polynomials have the same type as the polynomials of the input list.
In most cases, groebner::gbasis
computes
the basis via the Buchberger algorithm with the "sugar"
selection strategy being used.
For general information, see T. Becker and V. Weispfenning: "Gröbner Bases", Springer (1993). For details on the sugar selection strategy, see A. Giovini, T. Mora, G. Niesi, L. Robbiano, C. Traverso: "One sugar cube, please — or Selection strategies in the Buchberger algorithm", Proc. ISSAC '91, Bonn, 4954 (1991).