OpenMath

OpenMath is an extensible standard for representing the semantics of mathematical objects.

https://openmath.org

OpenMath Content Dictionary: polyu

Canonical URL:
http://www.openmath.org/cd/polyu.ocd
CD Base:
http://www.openmath.org/cd
CD File:
polyu.ocd
CD as XML Encoded OpenMath:
polyu.omcd
Defines:
poly_u_rep, polynomial_ring_u, polynomial_u, term
Date:
2004-03-30
Version:
2 (Revision 1)
Review Date:
2017-12-31
Status:
experimental

     This document is distributed in the hope that it will be useful, 
     but WITHOUT ANY WARRANTY; without even the implied warranty of 
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

     The copyright holder grants you permission to redistribute this 
     document freely as a verbatim copy. Furthermore, the copyright
     holder permits you to develop any derived work from this document
     provided that the following conditions are met.
       a) The derived work acknowledges the fact that it is derived from
          this document, and maintains a prominent reference in the 
          work to the original source.
       b) The fact that the derived work is not the original OpenMath 
          document is stated prominently in the derived work.  Moreover if
          both this document and the derived work are Content Dictionaries
          then the derived work must include a different CDName element,
          chosen so that it cannot be confused with any works adopted by
          the OpenMath Society.  In particular, if there is a Content 
          Dictionary Group whose name is, for example, `math' containing
          Content Dictionaries named `math1', `math2' etc., then you should 
          not name a derived Content Dictionary `mathN' where N is an integer.
          However you are free to name it `private_mathN' or some such.  This
          is because the names `mathN' may be used by the OpenMath Society
          for future extensions.
       c) The derived work is distributed under terms that allow the
          compilation of derived works, but keep paragraphs a) and b)
          intact.  The simplest way to do this is to distribute the derived
          work under the OpenMath license, but this is not a requirement.
     If you have questions about this license please contact the OpenMath
     society at http://www.openmath.org.

  Author: OpenMath Consortium (James Davenport)
  SourceURL: https://github.com/OpenMath/CDs

This CD contains operators to deal with polynomials and more precisely Univariate Polynomials. Note that recursive polynomials are regarded as univariates in their most significant variable (as defined by the order in PolynomialRingR: the first variable to appear is the most significant), with monomials in decreasing order of exponent, and coefficients being polynomials in the rest of the variables, and therefore univariates are a special case. This is provided as a separate CD to allow for univariate-only operations (e.g. composition) and for systems that only understand univariates, e.g. NTL. These polynomials are also used to express minimal polynomials for algebraic extensions (see setname2). 

Based on recursive polynomials 2003-08-06 JHD

     Definition of data-structure constructors

     The polynomial x^6 + 3*x^5 +2 can be conceptually encoded as
     poly_u_rep(x,
           term(6,1),
           term(5,3),
	   term(0,2))
     It lies in polynomial_ring_u(Z,x)

term

Role:
application
Description:

A constructor for monomials, that is products of powers and elements of the base ring. First argument is from N (the exponent of the variable implied by an outer poly_u_rep) second argument is a coefficient (from the ground field)

Signatures:
sts

[Next: poly_u_rep] [Last: polynomial_ring_u] [Top]

poly_u_rep

Role:
application
Description:

A constructor for the representation of polynomials. The first argument is the polynomial variable, the rest are monomials (in decreasing order of exponent).

Example:
The polynomial x^6 + 3*x^5 + 2 may be encoded as:
poly_u_rep ⁡ ( x , term ⁡ ( 6 , 1 ) , term ⁡ ( 5 , 3 ) , term ⁡ ( 0 , 2 ) )
Signatures:
sts

[Next: polynomial_u] [Previous: term] [Top]

polynomial_u

Role:
application
Description:

The constructor of Recursive Polynomials. The first argument is the polynomial ring containing the polynomial and the second is a "poly_u_rep".

Example:
The polynomial x^6 + 3*x^5 + 2 in the polynomial ring with the integers as the coefficient ring and variable x may be encoded as:
polynomial_u ⁡ ( polynomial_ring_u ⁡ ( Z , x ) , poly_u_rep ⁡ ( x , term ⁡ ( 6 , 1 ) , term ⁡ ( 5 , 3 ) , term ⁡ ( 0 , 2 ) ) )
Signatures:
sts

[Next: polynomial_ring_u] [Previous: poly_u_rep] [Top] 
    Polynomial ring constructor

polynomial_ring_u

Role:
application
Description:

The constructor of a univariate polynomial ring. The first argument is a ring (the ring of the coefficients), the second is the variable.

Commented Mathematical property (CMP):
Univariates are just recursive polynomials in one variable (though constructed using isomorphic, but different, constructors).
Formal Mathematical property (FMP):
polynomial_ring_u ⁡ ( R , x ) = polynomial_ring_r ⁡ ( R , x )
Example:
polynomial_ring_u ⁡ ( Z , x )
Signatures:
sts

[First: term] [Previous: polynomial_u] [Top]