SubresultantPolynomials
SubresultantPolynomials[poly1,poly2,var]
generates a list of subresultant polynomials of the polynomials poly1 and poly2 with respect to the variable var.
SubresultantPolynomials[poly1,poly2,var,Modulusp]
computes the subresultant polynomials modulo the prime p.
Details and Options
- SubresultantPolynomials require Exponent[poly1,var]≥Exponent[poly2,var].
- SubresultantPolynomials returns a list whose length is Exponent[poly2,var]+1.
- The first polynomial in the resulting list is Resultant[poly1,poly2,var].
Examples
open all close allBasic Examples (2)
This gives the list of subresultant polynomials of two univariate polynomials:
SubresultantPolynomials[(x - 1) ^ 2(x - 2)(x - 3), (x - 1)(x - 4) ^ 2, x]The list of subresultant polynomials of polynomials with symbolic coefficients:
SubresultantPolynomials[a x ^ 3 + b x ^ 2 + c x + d, 3 a x ^ 2 + b x + c, x]The first element is equal to Resultant of the input polynomials:
First[%] - Resultant[a x ^ 3 + b x ^ 2 + c x + d, 3 a x ^ 2 + b x + c, x]Scope (2)
Options (3)
Modulus (3)
By default, the subresultant polynomials are computed over the rational numbers:
SubresultantPolynomials[(x - 1) ^ 2(x - 2)(x - 3), (x - 1)(x - 4) ^ 2, x]Compute the subresultant polynomials of the same polynomials over the integers modulo 2:
SubresultantPolynomials[(x - 1) ^ 2(x - 2)(x - 3), (x - 1)(x - 4) ^ 2, x, Modulus -> 2]Compute the subresultant polynomials of the same polynomials over the integers modulo 5:
SubresultantPolynomials[(x - 1) ^ 2(x - 2)(x - 3), (x - 1)(x - 4) ^ 2, x, Modulus -> 5]Properties & Relations (2)
The degree of the 
subresultant polynomial is at most 
:
f = -14 + 4x + 3x ^ 2 + 2x ^ 3 + 5x ^ 8;
g = -21 + 7x + 6x ^ 2 + 5x ^ 3 + 3x ^ 7;sps = SubresultantPolynomials[f, g, x]Exponent[#, x]& /@ spsThe coefficient of the 
subresultant polynomial at 
is the 
principal subresultant coefficient:
psc = Table[Coefficient[sps[[i]], x, i - 1], {i, Length[sps]}]Subresultants computes the principal subresultant coefficients:
Subresultants[f, g, x]% - pscCoefficients of the subresultant polynomials are polynomials in the coefficients of the input:
SubresultantPolynomials[a x ^ 4 + b x + c, d x ^ 3 + e x + f, x]Possible Issues (1)
SubresultantPolynomials requires exact coefficients:
SubresultantPolynomials[x ^ 2 - 1.2, x - 3.4, x]
Text
Wolfram Research (2012), SubresultantPolynomials, Wolfram Language function, https://reference.wolfram.com/language/ref/SubresultantPolynomials.html.
CMS
Wolfram Language. 2012. "SubresultantPolynomials." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SubresultantPolynomials.html.
APA
Wolfram Language. (2012). SubresultantPolynomials. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SubresultantPolynomials.html