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


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  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


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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


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