# OpenMath Content Dictionary: polyd2

Canonical URL:
http://www.openmath.org/cd/polyd2.ocd
CD Base:
http://www.openmath.org/cd
CD File:
polyd2.ocd
CD as XML Encoded OpenMath:
polyd2.omcd
Defines:
Date:
2004-07-07
Version:
3
Review Date:
2006-04-01
Status:
experimental

This CD defines symbols for ordering of monomial for Distributed Multivariate Polynomials, which were defined in polyd1.

Original OpenMath v1.1 Poly 1997
Update to Current Format 1999-07-07 DPC
Move the names of rings to setname.ocd 1999-11-09 JHD
Delete those items moved to the new poly.ocd 1999-11-14 JHD
Delete those items pertaining to Groebner bases 2004-07-07 AMC

These are of use for canonical ways of writing polynomials and for Groebner bases


## ordering

Description:

Used as an attribute to indicate an ordering of the monomials in a polynomial or list of polynomials. The value of this attribute should be one of the constructors specifying ordering.

Signatures:
sts

 [Next: lexicographic] [Last: weighted_degree] [Top]
          The following orders on monomials have their standards definitions,
see, for example, "Ideals, Varieties and Algorithms", D. Cox,
J.B. Little and D. O'Shea, Springer Verlag.


## lexicographic

Description:

The lexicographic ordering of monomials.

Signatures:
sts

 [Next: reverse_lexicographic] [Previous: ordering] [Top]

## reverse_lexicographic

Description:

The reverse lexicographic ordering of monomials

Signatures:
sts

Description:

Total degree order, graded with the lexicographic ordering.

Signatures:
sts

Description:

Total degree order, graded with the reverse lexicographic ordering.

Signatures:
sts

## elimination

Description:

This is an ordering, which is partially in terms of one ordering, and partially in terms of another. First argument is a number of variables. Second is ordering to apply on the first so many variables. Third is an ordering on the rest, to be used to break ties.

Example:
$\mathrm{elimination}\left(1,\mathrm{lexicographic},\mathrm{graded_reverse_lexicographic}\right)$
Signatures:
sts

## matrix_ordering

Description:

The argument is a matrix with as many columns as indeterminates (= rank). Each row in turm is multiplied by the column vector of exponents to produce a weighting for comparison purposes.

Signatures:
sts

 [Next: weighted] [Previous: elimination] [Top]

## weighted

Description:

The first argument is a list of integers to act as variable weights, and the second is an ordering. The result is an ordering.

Signatures:
sts

 [Next: weighted_degree] [Previous: matrix_ordering] [Top]
  We need a few more orderings...

     Definition of some other constructors


## weighted_degree

Description:

The total degree of its argument, taking into account any weights declared. The value returned is an integer: non-negative if the weights are. We note that the degree of 0 is undefined.

Example:
$\mathrm{weighted_degree}\left(\mathrm{DMP}\left(\mathrm{poly_ring_d}\left(\mathbb{Q},3\right),\mathrm{SDMP}\left(\mathrm{term}\left(1,0,0,1\right),\mathrm{term}\left(2,2,0,0\right),\mathrm{term}\left(3,0,1,0\right),\mathrm{term}\left(4,1,0,0\right)\right)\right)\right)=3$
Signatures:
sts

 [First: ordering] [Previous: weighted] [Top]