UnivariatePolynomialCategory

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UnivariatePolynomialCategory

$\begin{usage} UnivariatePolynomialCategory~R: Category \end{usage}$

$\begin{params} {\em R} & \htmlref{\texttt{SumitType}}{SumitType} & The coeffic... ...main\\ & \htmlref{\texttt{ArithmeticType}}{ArithmeticType} &\\\end{params}$

$\begin{descr} UnivariatePolynomialCategory~is the category of univariate polyno... ...ary domain R and with respect to the power basis $(x^n)_{n \ge 0}$. \end{descr}$

Exports

`UnivariatePolynomialAlgebra` R
`apply`:	(%, R) $\to$ R	Evaluate a polynomial
`apply`:	(%, %) $\to$ %	Evaluate a polynomial
`equal?`:	(%, %, %, `Integer`) $\to$ `Boolean`	Truncated equality
`Horner`:	(%, R) $\to$ (%, R)	Horner division by

if has CommutativeRing then
DifferentialRing

lift: (Derivation R, %) $\to$ Derivation % Extend a derivation

if has CommutativeRing and has QRing then
integrate: % $\to$ % Integration

(%, Integer) $\to$ %

if has FactorizationRing then

`factor`:	% $\to$ (R, `Product` %)	Factorisation into irreducibles
`fractionalRoots`:	% $\to$ `Generator` `FractionalRoot` R	Roots in the fraction field
`roots`:	% $\to$ `Generator` `FractionalRoot` R	Roots in the coefficient ring

if has Field then
EuclideanDomain

sparseMultiple: (%, Integer) $\to$ % Multiple in

if has GcdDomain then
DecomposableRing

GcdDomain

squareFree: % $\to$ (R, Product %) Squarefree factorisation

squareFreePart: % $\to$ % Squarefree part

if has GcdDomain and has RationalRootRing then
dispersion: % $\to$ Integer Dispersion

(%, %) $\to$ Integer

integerDistances: % $\to$ List Integer Integer spread

(%, %) $\to$ List Integer

universalBound: (%,%) $\to$ List Cross(%, Integer) Universal bound

if has IntegralDomain then

`IntegralDomain`
`monicDivide`:	(%, %) $\to$ (%, %)	Polynomial division
`monicRemainder`:	(%, %) $\to$ %	Remainder
`monicRemainder!`:	(%, %) $\to$ %	Remainder
`pseudoDivide`:	(%, %) $\to$ (%, %)	Polynomial pseudo-division
`pseudoRemainder`:	(%, %) $\to$ %	Pseudo-remainder
`pseudoRemainder!`:	(%, %) $\to$ %	Pseudo-remainder
`resultant`:	(%, %) $\to$ R	Resultant of 2 polynomials

if has OrderedArithmeticType then
height: % $\to$ R Max norm over all the coefficients

if has RationalRootRing then
RationalRootRing

integerRoots: % $\to$ Generator FractionalRoot Integer Integer roots

rationalRoots: % $\to$ Generator FractionalRoot Integer Rational roots

if has Ring then
ordinaryPoint: % $\to$ Integer Point where a polynomial is nonzero

values: (%, R) $\to$ Generator R Generate values of a polynomial

if has Specializable then
Specializable

Subsections

Next: apply Up: Univariate Polynomials and Series Previous: compose,translate Contents Index

Manuel Bronstein 2000-12-13

`DifferentialRing`
`lift`:	(`Derivation` R, %) $\to$ `Derivation` %	Extend a derivation

`EuclideanDomain`
`sparseMultiple`:	(%, `Integer`) $\to$ %	Multiple in

`DecomposableRing`
`GcdDomain`
`squareFree`:	% $\to$ (R, `Product` %)	Squarefree factorisation
`squareFreePart`:	% $\to$ %	Squarefree part

`dispersion`:	% $\to$ `Integer`	Dispersion
	(%, %) $\to$ `Integer`
`integerDistances`:	% $\to$ `List` `Integer`	Integer spread
	(%, %) $\to$ `List` `Integer`
`universalBound`:	(%,%) $\to$ `List` `Cross`(%, `Integer`)	Universal bound

`RationalRootRing`
`integerRoots`:	% $\to$ `Generator` `FractionalRoot` `Integer`	Integer roots
`rationalRoots`:	% $\to$ `Generator` `FractionalRoot` `Integer`	Rational roots

`ordinaryPoint`:	% $\to$ `Integer`	Point where a polynomial is nonzero
`values`:	(%, R) $\to$ `Generator` R	Generate values of a polynomial