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:

Introduced in 1999
 (4.0)
 |
Updated in 2012
 (9.0)