# OpenMath Content Dictionary: polyd3

Canonical URL:
http://www.openmath.org/cd/polyd3.ocd
CD Base:
http://www.openmath.org/cd
CD File:
polyd3.ocd
CD as XML Encoded OpenMath:
polyd3.omcd
Defines:
collect, list_to_poly_d, poly_d_named_to_arith, poly_d_to_arith
Date:
2004-07-07
Version:
1
Review Date:
2006-04-01
Status:
experimental

This CD contains conversions between different data structures for polynomials.

first attempt


## collect

Description:

This a binary function. Its first argument should be a DMP f, its second argument a list of positive integers L. When applied to f and L, it represents the DMP with coefficients from the poly_ring_d whose variables only have indices i for i not occurring in the list L, and whose monomials are built up from the variables indexed by the entries of L.

Signatures:
sts

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## list_to_poly_d

Description:

This symbol is a function with two arguments. The first argument is a ring R and the second argument is a list L. The entries of L are elements of R or can be cast canconically onto elements of R. When applied to R and L, the symbol denotes the distributed (univariate) polynomial over R with terms (L[i-1],i) for i running over the indices of L (i=1, ..., length(L)).

Example:
The polynomial X^2+4X+7 with integers coefficients can be represented, without recourse to X, by
$\mathrm{list_to_poly_d}\left(Z,\left(7,4,1\right)\right)$
Thus, the above expression is semantically equivalent to
$\mathrm{DMP}\left(Z,\mathrm{SDMP}\left(\mathrm{term}\left(7,0\right),\mathrm{term}\left(4,1\right),\mathrm{term}\left(1,2\right)\right)\right)$
Signatures:
sts

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## poly_d_to_arith

Description:

This symbol is a binary function. The first argument is a DMP and the second argument is a list of objects, typically variables or arithmetic expressions, at least as many as there are variables in the ring to which the DMP belongs. When applied to R and L, the symbol denotes the arithmetic expression that is the sum of the terms with the i-th variable of the ring of the DMP being substituted by the i-th expression or variable of the list L.

This expression is very close to a substitute, or an application of the DMP, as
viewed as a function, to
as many arguments as there are variables.

Example:
The polynomial X^2+4X+7 with integer coefficients has poly_d form
$\mathrm{poly_d_to_arith}\left(\mathrm{DMP}\left(\mathrm{poly_ring_d}\left(Z,1\right),\mathrm{SDMP}\left(\mathrm{term}\left(7,0\right),\mathrm{term}\left(4,1\right),\mathrm{term}\left(1,2\right)\right)\right),\left(X\right)\right)$
This expression represents
$7+4X+{X}^{2}$
Signatures:
sts

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## poly_d_named_to_arith

Description:

This symbol is a unary function. Its argument is a DMP with named variables. When applied to R, the symbol denotes the arithmetic expression that is the sum of the terms.

Example:
The polynomial X^2+4X+7 with integers coefficients has poly_d_named form
$\mathrm{poly_d_to_arith}\left(\mathrm{DMP}\left(\mathrm{poly_ring_d_named}\left(Z,X\right),\mathrm{SDMP}\left(\mathrm{term}\left(7,0\right),\mathrm{term}\left(4,1\right),\mathrm{term}\left(1,2\right)\right)\right)\right)$
This expression represents
$7+4X+{X}^{2}$
Signatures:
sts

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