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:
elimination, graded_lexicographic, graded_reverse_lexicographic, lexicographic, matrix_ordering, ordering, reverse_lexicographic, weighted, weighted_degree
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


[Next: graded_lexicographic] [Previous: lexicographic] [Top]

graded_lexicographic

Description:

Total degree order, graded with the lexicographic ordering.

Signatures:
sts


[Next: graded_reverse_lexicographic] [Previous: reverse_lexicographic] [Top]

graded_reverse_lexicographic

Description:

Total degree order, graded with the reverse lexicographic ordering.

Signatures:
sts


[Next: elimination] [Previous: graded_lexicographic] [Top]

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:
elimination ( 1 , lexicographic , graded_reverse_lexicographic )
Signatures:
sts


[Next: matrix_ordering] [Previous: graded_reverse_lexicographic] [Top]

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:
weighted_degree ( DMP ( poly_ring_d ( Q , 3 ) , SDMP ( term ( 1 , 0 , 0 , 1 ) , term ( 2 , 2 , 0 , 0 ) , term ( 3 , 0 , 1 , 0 ) , term ( 4 , 1 , 0 , 0 ) ) ) ) = 3
Signatures:
sts


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