# OpenMath Content Dictionary: polynomial3

Canonical URL:
http://www.openmath.org/cd/polynomial3.ocd
CD Base:
http://www.openmath.org/cd
CD File:
polynomial3.ocd
CD as XML Encoded OpenMath:
polynomial3.omcd
Defines:
factors, gcd, lcm, quotient, remainder
Date:
2004-07-12
Version:
0
Review Date:
2006-07-12
Status:
experimental

This CD holds a collection of basic modular arithmetic for polynomials over fields. The data structures for polynomials can be arithmetic expressions, for instance using the ring1.expression symbol, or DMP as in the CD polyd1.

## gcd

Description:

The n-ary greatest common divisor for univariate polynomials over fields.

Example:
The gcd(X,Y,Z).
$\mathrm{gcd}\left(X,Y,Z\right)$
Signatures:
sts

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## lcm

Role:
application
Description:

The least common multiple of its polynomial arguments. This is unique up to units, but the choice must be compatible with that made for gcd, so that the product of the gcd and the lcm equals the product of all arguments.

Commented Mathematical property (CMP):
for all x,y we have lcm(x,y) = (x*y)/gcd(x,y)
Formal Mathematical property (FMP):
$\forall x,y.\mathrm{lcm}\left(x,y\right)=\frac{xy}{\mathrm{gcd}\left(x,y\right)}$
Signatures:
sts

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## factors

Description:

This symbol is a unary function, whose argument should be a polynomial f. When applied to f, it represents a complete list of irreducible factors of f.

Example:
The following expression represents the list [X+1,X+1] of rational polynomials.
$\mathrm{factors}\left(\mathrm{expression}\left(\mathrm{expression}\left(\mathrm{poly_ring}\left(Q,X\right),{X}^{2}+2X+1\right)\right)\right)$
Signatures:
sts

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## quotient

Role:
application
Description:

This symbol represents the binary division operator on univariate polynomials over fields. That is, for univariate polynomials a and b, quotient(a,b) denotes the polynomial q such that a=b*q+r, with degree(r) less than degree(b).

Commented Mathematical property (CMP):
For all a,b with a,b univariate polynomials over a field F we have a = b * quotient(a,b) + remainder(a,b) and degree(remainder(a,b)) is less than degree(b).
Formal Mathematical property (FMP):
$\forall a,b.a\in \mathrm{poly_ring_d}\left(F,1\right)\wedge b\in \mathrm{poly_ring_d}\left(F,1\right)⇒a=b\mathrm{quotient}\left(a,b\right)+\mathrm{remainder}\left(a,b\right)\wedge \mathrm{degree}\left(\right)<\mathrm{degree}\left(b\right)$
Signatures:
sts

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## remainder

Role:
application
Description:

The symbol represents a binary function, whose arguments should be univariate polynomials in the same polynomial ring whose coefficient ring is a field. When applied to a and b, it represents the polynomial remainder after division of a by b.

For univariate polynomials a and b, remainder(a,b) denotes r such that a=b*q+r, with degree(r) less
than degree(b).
See the definition of the symbol quotient for a formal statement of this property.

Signatures:
sts

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