# OpenMath Content Dictionary: polyr

Canonical URL:
http://www.openmath.org/cd/polyd.ocd
CD Base:
http://www.openmath.org/cd
CD File:
polyr.ocd
CD as XML Encoded OpenMath:
polyr.omcd
Defines:
poly_r_rep, polynomial_r, polynomial_ring_r, term
Date:
2004-03-30
Version:
3 (Revision 1)
Review Date:
2017-12-31
Status:
experimental

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Author: OpenMath Consortium
SourceURL: https://github.com/OpenMath/CDs

This CD contains operators to deal with polynomials and more precisely Recursive Polynomials. Note that 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. This means that polynomials have a unique representation, except for the fact that yz \in Z[x,y,z] could also be represented as x^0yz. This latter is discouraged, but currently not expressly forbidden.

Original OpenMath v1.1 Poly 1997
Update to Current Format 1999-07-07 DPC
Move the names of rings to setname1.ocd 1999-11-09 JHD
Delete those items moved to the new poly.ocd 1999-11-14 JHD
Convert to recursive polynomials 1999-11-20 JHD
Definition of data-structure constructors
The polynomial x^2*y^6 + 3*y^5 can be conceptually encoded as
poly_r_rep(x,
term(2,poly_r_rep(y,
term(6,1))),
term(0,poly_r_rep(y,
term(5,3))))
It lies in polynomial_ring_r(Z,x,y) (and other rings, of course)

The polynomial 2*y^3*z^5 + x + 1 can be conceptually encoded as
poly_r_rep(x,
term(1,1),
term(0,poly_r_rep(y,
term(3,poly_r_rep(z,
term(5,2))),
term(0,1))))

## 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_r_rep) second argument is a coefficient (from the ground field, or a polynomial in lesser variables).

Signatures:
sts

 [Next: poly_r_rep] [Last: polynomial_ring_r] [Top]

## poly_r_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^2*y^6 + 3*x^0*y^5 = x^2*y^6 + 3*y^5 may be encoded as:
$\mathrm{poly_r_rep}\left(x,\mathrm{term}\left(2,\mathrm{poly_r_rep}\left(y,\mathrm{term}\left(6,1\right)\right)\right),\mathrm{term}\left(0,\mathrm{poly_r_rep}\left(y,\mathrm{term}\left(5,3\right)\right)\right)\right)$
Signatures:
sts

 [Next: polynomial_r] [Previous: term] [Top]

## polynomial_r

Role:
application
Description:

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

Example:
The polynomial x^2*y^6 + 3*x^0*y^5 = x^2*y^6 + 3*y^5 in the polynomial ring with the integers as the coefficient ring and variables x,y in that order may be encoded as:
$\mathrm{polynomial_r}\left(\mathrm{polynomial_ring_r}\left(\mathbb{Z},x,y\right),\mathrm{poly_r_rep}\left(x,\mathrm{term}\left(2,\mathrm{poly_r_rep}\left(y,\mathrm{term}\left(6,1\right)\right)\right),\mathrm{term}\left(0,\mathrm{poly_r_rep}\left(y,\mathrm{term}\left(5,3\right)\right)\right)\right)\right)$
Signatures:
sts

 [Next: polynomial_ring_r] [Previous: poly_r_rep] [Top]
Polynomial ring constructor

## polynomial_ring_r

Role:
application
Description:

The constructor of a recursive polynomial ring. The first argument is a ring (the ring of the coefficients), the rest are the variables (in order).

Example:
$\mathrm{polynomial_ring_r}\left(\mathbb{Z},x,y\right)$
Signatures:
sts

 [First: term] [Previous: polynomial_r] [Top]