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

open allclose all

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:

Introduced in 2012
 (9.0)