# Subresultants

Subresultants[poly1,poly2,var]

generates a list of the principal subresultant coefficients of the polynomials poly1 and poly2 with respect to the variable var.

Subresultants[poly1,poly2,var,Modulusp]

computes the principal subresultant coefficients modulo the prime p.

# Details and Options

• The first k subresultants of two polynomials a and b, both with leading coefficient one, are zero when a and b have k common roots.
• Subresultants returns a list whose length is Min[Exponent[poly1,var],Exponent[poly2,var]]+1. »

# Examples

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## Basic Examples(2)

The first three principal subresultant coefficients (PSCs) are zero when there are three common roots, multiplicities counted:

PSCs of two cubic polynomials:

When the polynomials have a pair of equal roots, the first PSC disappears:

When two pairs of roots are equal, the first two PSCs disappear:

## Scope(2)

Principal subresultant coefficients of univariate polynomials are numbers:

Principal subresultant coefficients are polynomials in the coefficients of input polynomials:

## Options(3)

### Modulus(3)

By default, the principal subresultant coefficients are computed over the rational numbers:

Compute the principal subresultant coefficients over the integers modulo 2:

Compute the principal subresultant coefficients over the integers modulo 7:

## Applications(2)

Find conditions for two polynomials to have exactly two common roots:

Check that for the first solution f and g have exactly two common roots:

Find conditions for a quartic to have exactly two distinct roots:

Check that for the first solution f has exactly two distinct roots:

## Properties & Relations(3)

Multiplicity of roots counts in determining the number of zero subresultants:

The length is determined by the minimum polynomial degree:

The first element of Subresultants is equal to Resultant:

Wolfram Research (1999), Subresultants, Wolfram Language function, https://reference.wolfram.com/language/ref/Subresultants.html (updated 2022).

#### Text

Wolfram Research (1999), Subresultants, Wolfram Language function, https://reference.wolfram.com/language/ref/Subresultants.html (updated 2022).

#### CMS

Wolfram Language. 1999. "Subresultants." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Subresultants.html.

#### APA

Wolfram Language. (1999). Subresultants. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Subresultants.html

#### BibTeX

@misc{reference.wolfram_2024_subresultants, author="Wolfram Research", title="{Subresultants}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Subresultants.html}", note=[Accessed: 24-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_subresultants, organization={Wolfram Research}, title={Subresultants}, year={2022}, url={https://reference.wolfram.com/language/ref/Subresultants.html}, note=[Accessed: 24-July-2024 ]}