Resultant

Resultant[poly1,poly2,var]

computes the resultant of the polynomials poly1 and poly2 with respect to the variable var.

Resultant[poly1,poly2,var,Modulusp]

computes the resultant modulo the prime p.

Details and Options

  • The resultant of two polynomials p and q, both with leading coefficient 1, is the product of all the differences pi-qj between roots of the polynomials. The resultant is always a number or a polynomial.

Examples

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Basic Examples  (1)

The resultant vanishes exactly when the polynomials have roots in common:

Scope  (4)

Resultant of polynomials with numeric coefficients:

Resultant of polynomials with parametric coefficients:

Resultant over integers modulo 3:

The resultant reflects the multiplicities of roots:

Generalizations & Extensions  (1)

The resultant of rational functions is defined using the multiplicative property:

Options  (4)

Method  (1)

This compares timings of the available methods of resultant computation:

Modulus  (3)

By default the resultant is computed over the rational numbers:

Compute the resultant of the same polynomials over the integers modulo 2:

Compute the resultant of the same polynomials over the integers modulo 3:

Applications  (2)

Decide whether two polynomials have common roots:

Find conditions for two polynomials to have common roots:

Properties & Relations  (6)

The resultant is zero if and only if the polynomials have a common root:

The polynomials have a zero resultant if and only if they have a nonconstant PolynomialGCD:

The resultant can be represented in terms of roots as :

Equation relates Discriminant and Resultant:

GroebnerBasis can also be used to find conditions for common roots:

The same problem can also be solved using Reduce, Resolve, and Eliminate:

Possible Issues  (1)

The following two polynomials have no common root:

Using approximate coefficients they will appear to have a common root:

Using higher precision shows they have no common root:

Wolfram Research (1988), Resultant, Wolfram Language function, https://reference.wolfram.com/language/ref/Resultant.html.

Text

Wolfram Research (1988), Resultant, Wolfram Language function, https://reference.wolfram.com/language/ref/Resultant.html.

BibTeX

@misc{reference.wolfram_2020_resultant, author="Wolfram Research", title="{Resultant}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/Resultant.html}", note=[Accessed: 15-January-2021 ]}

BibLaTeX

@online{reference.wolfram_2020_resultant, organization={Wolfram Research}, title={Resultant}, year={1988}, url={https://reference.wolfram.com/language/ref/Resultant.html}, note=[Accessed: 15-January-2021 ]}

CMS

Wolfram Language. 1988. "Resultant." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Resultant.html.

APA

Wolfram Language. (1988). Resultant. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Resultant.html