# 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: 