Package 'resultant'

Greatest common divisor of two polynomials

Description

Greatest common divisor of two polynomials with rational coefficients.

Usage

gcd(qspray1, qspray2, utcf = FALSE)

Arguments

qspray1, qspray2	two qspray polynomials with at more nine variables
utcf	Boolean, whether to get the greatest common divisor up to a constant factor (this can be faster)

Value

A qspray polynomial.

Examples

library(resultant)
x <- qlone(1)
y <- qlone(2)
g <- x^2 + 2*x*y + 1
p <- g * (y^2 + x^2)
q <- g * (y + x^3 + 2)
gcd(p, q)

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Integral division of two polynomials

Description

Integral division (division without remainder) of two polynomials with rational coefficients.

Usage

integralDivision(qspray1, qspray2, check = TRUE)

Arguments

qspray1, qspray2	two qspray polynomials having at most nine variables
check	Boolean, whether to check that qspray2 divides qspray1

Value

If check=TRUE, this returns NULL if qspray2 does not divide qspray1, otherwise this returns a qspray polynomial, the quotient of qspray1 by qspray2. If check=FALSE, this always returns a qspray polynomial, which is the quotient of qspray1 by qspray2 if qspray2 divides qspray1, otherwise it is an undefined polynomial. So you can use check=FALSE only when you are sure that qspray2 divides qspray1.

See Also

univariateDivision, qsprayDivision.

Examples

library(resultant)
x <- qlone(1)
y <- qlone(2)
q <- x^2 + 2*x*y + 1
qspray1 <- q * (x^4 + y^2 + 2)
qspray2 <- x^4 + y^2 + 2
integralDivision(qspray1, qspray2) == q # should be TRUE

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Number of real roots

Description

Number of distinct real roots of a univariate polynomial.

Usage

numberOfRealRoots(qspray)

Arguments

qspray

a univariate qspray polynomial

Value

An integer, the number of real roots of the polynomial.

Note

The roots are not counted with their multiplicity.

Examples

library(resultant)
x <- qlone(1)
P <- 2*x^4 + x^3 - 3*x^2 - x + 1
numberOfRealRoots(P)

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Number of real roots in an interval

Description

Number of distinct real roots of a univariate polynomial in a given interval.

Usage

numberOfRealRootsInInterval(qspray, lower, upper, closed = TRUE)

Arguments

qspray	a univariate qspray polynomial
lower, upper	the bounds of the interval, bigq numbers or objects coercible to bigq numbers, and it is also possible to set lower = -Inf and upper = Inf
closed	Boolean, whether to consider the interval is closed or open

Value

An integer, the number of real roots of the polynomial in the interval.

Note

The roots are not counted with their multiplicity.

Examples

library(resultant)
x <- qlone(1)
P <- 2*x^4 + x^3 - 3*x^2 - x + 1
numberOfRealRootsInInterval(P, 0, 1)

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Principal Sturm-Habicht sequence of a polynomial

Description

Principal Sturm-Habicht sequence of a polynomial with rational coefficients.

Usage

principalSturmHabicht(qspray, var = 1)

Arguments

qspray	a qspray polynomial with at most nine variables
var	integer indicating with respect to which variable the resultant is desired (e.g. 1 for x and 2 for y)

Value

For a univariate polynomial, this returns a vector of bigq rational numbers. For a multivariate polynomial, this returns a list of qspray polynomials that do not involve the var-th variable.

Examples

library(resultant)
x <- qlone(1)
y <- qlone(2)
qspray <- x^3*y^2 - 5*x*y^2 + 7*x - 2
principalSturmHabicht(qspray, var = 1)
principalSturmHabicht(qspray, var = 2)

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Principal subresultants of two polynomials

Description

Principal subresultants of two polynomials with rational coefficients.

Usage

principalSubresultants(qspray1, qspray2, var = 1)

Arguments

qspray1, qspray2	two qspray polynomials with at most nine variables
var	integer indicating with respect to which variable the subresultants are desired (e.g. 1 for x and 2 for y)

Value

If both qspray1 and qspray2 are univariate polynomials, the function returns a vector of bigq rational numbers. Otherwise, it returns a list of qspray polynomials that do not involve the var-th variable.

Examples

library(resultant)
x <- qlone(1)
y <- qlone(2)
p <- x^2 * y * (y^2 - 5*x + 6)
q <- x^2 * y * (3*y + 2)
principalSubresultants(p, q, var = 1) # should be 0, 0, non-zero, ...
principalSubresultants(p, q, var = 2) # should be 0, non-zero, ...

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Resultant of two polynomials

Description

Resultant of two polynomials with rational coefficients.

Usage

resultant(qspray1, qspray2, var = 1)

Arguments

qspray1, qspray2	two qspray polynomials with at most nine variables
var	integer indicating with respect to which variable the resultant is desired (e.g. 1 for x and 2 for y)

Value

If both qspray1 and qspray2 are univariate polynomials, the function returns a bigq rational number. Otherwise, it returns a qspray polynomial that does not involve the var-th variable.

Examples

library(resultant)
x <- qlone(1)
y <- qlone(2)
f <- x^4 - x^3 + x^2 - 2*x*y^2 + y^4
g <- x - 2*y^2
resultant(f, g, var = 1)
resultant(f, g, var = 2)

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Square-free factorization of a polynomial

Description

Square-free factorization of a polynomial with rational coefficients.

Usage

squareFreeFactorization(qspray)

Arguments

qspray

a qspray polynomial having at most nine variables

Value

A list with two fields constantFactor and nonConstantFactors. In the field constantFactor, there is a bigq rational number, the constant factor of the factorization. In the field nonConstantFactors, there is a list providing the square-free and pairwise coprime qspray polynomials of the factorization with their multiplicity.

Examples

library(resultant)
x <- qlone(1)
y <- qlone(2)
p <- x^8*y^7 + 2*x^7*y^5 + x^6*y^4 + 2*x^5*y^2
squareFreeFactorization(p)

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Sturm-Habicht sequence of a polynomial

Description

Sturm-Habicht sequence of a polynomial with rational coefficients.

Usage

SturmHabicht(qspray, var = 1)

Arguments

qspray	a qspray polynomial having at most nine variables
var	index of the variable with respect to which the Sturm-Habicht sequence will be computed

Value

A list of qspray polynomials, the Sturm-Habicht sequence of qspray, starting with the 0-th Sturm-Habicht polynomial.

Examples

library(resultant)
x <- qlone(1)
y <- qlone(2)
SturmHabicht(x^3*y^2 + 2*x*y + 1)

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Subresultants of two polynomials

Description

Subresultants of two polynomials with rational coefficients.

Usage

subresultants(qspray1, qspray2, var = 1)

Arguments

qspray1, qspray2	two qspray polynomials having at most nine variables
var	integer indicating with respect to which variable the subresultants will be computed (e.g. 1 for x and 2 for y)

Value

A list of qspray polynomials.

Examples

library(resultant)
x <- qlone(1)
y <- qlone(2)
p <- x^2 * y * (y^2 - 5*x + 6)
q <- x^2 * y * (3*y + 2)
subresultants(p, q, var = 1)
subresultants(p, q, var = 2)

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Division of univariate polynomials

Description

Division with remainder of univariate polynomials with rational coefficients.

Usage

univariateDivision(qspray1, qspray2)

Arguments

qspray1, qspray2

two univariate qspray polynomials

Value

A list of two univariate qspray polynomials, the quotient of the division in the field Q of the list, and the remainder in the field R.

See Also

integralDivision, qsprayDivision.

Examples

library(resultant)
x <- qlone(1)
qspray1 <- 2*x^4 + x^3 - 3*x^2 - x + 1
qspray2 <- x^2 - 5*x + 10
division <- univariateDivision(qspray1, qspray2)
Q <- division[["Q"]]; R <- division[["R"]]
qspray1 == Q*qspray2 + R # should be TRUE

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