BUILT-IN MATHEMATICA SYMBOL

Subresultants

generates a list of the principal subresultant coefficients of the polynomials

and

with respect to the variable var.

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. »

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:

The first three principal subresultant coefficients (PSCs) are zero when there are three common roots, multiplicities counted:

Out[1]=

PSCs of two cubic polynomials:

Out[1]=

When the polynomials have a pair of equal roots, the first PSC disappears:

Out[2]=

When two pairs of roots are equal, the first two PSCs disappear:

Out[3]=

Principal subresultant coefficients of univariate polynomials are numbers:

Principal subresultant coefficients are polynomials in the coefficients of input polynomials:

Find conditions for two polynomials to have exactly two common roots:

Check that for the first solution

and

have exactly two common roots:

Find conditions for a quartic to have exactly two distinct roots:

Check that for the first solution

has exactly two distinct roots:

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: