SubresultantPolynomialRemainders

SubresultantPolynomialRemainders[poly1,poly2,var]

gives the subresultant polynomial remainder sequence of the polynomials poly1 and poly2 with respect to the variable var.

SubresultantPolynomialRemainders[poly1,poly2,var,Modulusp]

computes the subresultant polynomial remainder sequence modulo the prime p.

Details and Options

  • SubresultantPolynomialRemainders is also known as subresultant polynomial remainder sequence or prs.
  • SubresultantPolynomialRemainders gives a list of polynomials of decreasing degrees in var.
  • Each polynomial in the list is a constant multiple of the PolynomialRemainder of the previous two polynomials, with poly1 and poly2 being the first two elements.
  • The last polynomial in the resulting list is a constant multiple of the polynomial GCD of univariate polynomials poly1 and poly2 in the variable var.

Examples

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Basic Examples  (2)

This gives the subresultant polynomial remainder sequence of two polynomials:

Subresultant polynomial remainder sequence of polynomials with symbolic coefficients:

The last element differs from the GCD of the polynomials by a factor independent of :

Scope  (2)

SubresultantPolynomialRemainders gives a list of polynomials of decreasing degrees:

Coefficients of the resulting polynomials are polynomials in the coefficients of the input:

Options  (3)

Modulus  (3)

By default, the subresultant prs is computed over the rational numbers:

Compute the subresultant prs of the same polynomials over the integers modulo 2:

Compute the subresultant prs of the same polynomials over the integers modulo 3:

Properties & Relations  (3)

The first two elements of the subresultant prs are the input polynomials:

The remaining elements are polynomial remainders, except for a constant factor:

All elements of the subresultant prs are divisible by the PolynomialGCD of the input polynomials:

The elements from prs, except initial polynomials, are a subset of SubresultantPolynomials:

Possible Issues  (1)

SubresultantPolynomialRemainders requires exact coefficients:

Wolfram Research (2012), SubresultantPolynomialRemainders, Wolfram Language function, https://reference.wolfram.com/language/ref/SubresultantPolynomialRemainders.html.

Text

Wolfram Research (2012), SubresultantPolynomialRemainders, Wolfram Language function, https://reference.wolfram.com/language/ref/SubresultantPolynomialRemainders.html.

CMS

Wolfram Language. 2012. "SubresultantPolynomialRemainders." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SubresultantPolynomialRemainders.html.

APA

Wolfram Language. (2012). SubresultantPolynomialRemainders. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SubresultantPolynomialRemainders.html

BibTeX

@misc{reference.wolfram_2024_subresultantpolynomialremainders, author="Wolfram Research", title="{SubresultantPolynomialRemainders}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/SubresultantPolynomialRemainders.html}", note=[Accessed: 16-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_subresultantpolynomialremainders, organization={Wolfram Research}, title={SubresultantPolynomialRemainders}, year={2012}, url={https://reference.wolfram.com/language/ref/SubresultantPolynomialRemainders.html}, note=[Accessed: 16-November-2024 ]}