Discriminant

Discriminant[poly,var]

computes the discriminant of the polynomial poly with respect to the variable var.

Discriminant[poly,var,Modulusp]

computes the discriminant modulo .

Details and Options

  • The discriminant of a polynomial with leading coefficient one is the product over all pairs of roots , of .
  • A Method option can be given, with typical possible values being Automatic, "SylvesterMatrix", "BezoutMatrix", "Subresultants", and "Modular".

Examples

open allclose all

Basic Examples  (1)

Discriminant of a quadratic:

Scope  (7)

Discriminant of a polynomial with numeric coefficients:

Discriminant of a general cubic:

Discriminant of a general quintic:

Discriminants are squares of differences of roots:

Discriminant over integers modulo 3:

Discriminant over a finite field:

Compute the discriminant of a polynomial of degree :

Options  (4)

Method  (1)

This compares timings of the available methods of discriminant computation:

Modulus  (3)

By default the discriminant is computed over the rational numbers:

Compute the discriminant of the same polynomial over the integers modulo 2:

Compute the discriminant of the same polynomial over the integers modulo 3:

Applications  (2)

Decide whether a polynomial has multiple roots:

Find the condition for a cubic to have multiple roots:

Properties & Relations  (3)

The discriminant is zero if and only if the polynomial has multiple roots:

The discriminant can be represented in terms of roots as :

Equation relates Discriminant and Resultant:

Possible Issues  (1)

Using exact coefficients, this indicates no common root:

With approximate coefficients, this does indicate a common root:

in this case, using higher precision resolves the problem:

Wolfram Research (2007), Discriminant, Wolfram Language function, https://reference.wolfram.com/language/ref/Discriminant.html (updated 2023).

Text

Wolfram Research (2007), Discriminant, Wolfram Language function, https://reference.wolfram.com/language/ref/Discriminant.html (updated 2023).

CMS

Wolfram Language. 2007. "Discriminant." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/Discriminant.html.

APA

Wolfram Language. (2007). Discriminant. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Discriminant.html

BibTeX

@misc{reference.wolfram_2024_discriminant, author="Wolfram Research", title="{Discriminant}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/Discriminant.html}", note=[Accessed: 15-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_discriminant, organization={Wolfram Research}, title={Discriminant}, year={2023}, url={https://reference.wolfram.com/language/ref/Discriminant.html}, note=[Accessed: 15-November-2024 ]}