Coefficient Domain

GRÖBNER is capable of computing Gröbner bases for sets of polynomials that are subsets of , for various fields . is called the coefficient domain, are called the ring variables and have to be entered at run-time.

If Coefficient Domain is chosen from the general setup menu the actual setting for can be chosen from the following menu:

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Current setting is: rational_numbers
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Rational Numbers ........ RN
Rational Functions ...... RF
Finite Field Zp ......... FF
Galois Field ............ GF
Floting Point Numbers ... FPN
Leave Unmodified ........ <return>

First, the current setting of the coefficient domain is displayed, the following choices are possible then:

RN

chooses rational numbers as coefficient domain, i.e. .

RF

chooses rational functions as coefficient domain, i.e. . is called the rational function base domain and can be chosen from an immediately following menu:

RN

chooses ,

FF

chooses . p is called the characteristic of the finite field and must be entered at run-time.

are called the field variables and have to be entered at run-time.

FF

chooses finite fields of the form as coefficient domain, i.e. . p is called the characteristic of the finite field and must be entered at run-time.

GF

chooses finite fields of the form as coefficient domain, i.e. . p is called the characteristic of the finite field and n is called the degree of the field extension. Both p and n have to be entered at run-time.

is isomorphic to , where q is an irreducible polynomial over of degree n. This isomorphism is used to represent elements of as univariate polynomials over of degree less than n. The actual name of the variable x has to be entered at run-time.

In order to perform arithmetic in , p and q are required. At run-time p has to be entered first. In case p=2 the degree of the field extension, n, has to be entered. For n<10 irreducible polynomials over of degree n are predefined. For or an irreducible polynomial over of degree n must be supplied at run-time.

Although is a special case of , namely n=1, we implemented finite fields of the form seperately, since arithmetic can be done more efficiently.

FPN

Chooses floating point numbers as coefficient domain, i.e. . This is an experimental implementaion of floating point coefficients.

Note: The menu for setting up the coefficient domain can also be accessed directly by calling the program coef_setup.

See Section 2.1 for the influence of the chosen coefficient domain.