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Subresultants

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.
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:
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PSCs of two cubic polynomials:
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When the polynomials have a pair of equal roots, the first PSC disappears:
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When two pairs of roots are equal, the first two PSCs disappear:
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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:
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