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
open all close allBasic Examples (2)
The first three principal subresultant coefficients (PSCs) are zero when there are three common roots, multiplicities counted:
Subresultants[(x - 1)(x - 2) ^ 2(x - 3) ^ 3, (x - 1)(x - 2) ^ 2(x - 4)(x - 5) ^ 2, x]PSCs of two cubic polynomials:
psc = Subresultants[(x - a)(x - b)(x - c), (x - 1)(x - 2)(x - 3), x]When the polynomials have a pair of equal roots, the first PSC disappears:
psc /. {a -> 1}When two pairs of roots are equal, the first two PSCs disappear:
psc /. {a -> 1, b -> 2}Scope (2)
Principal subresultant coefficients of univariate polynomials are numbers:
Subresultants[2x ^ 7 + 3x ^ 3 - 7x + 1, 3x ^ 5 - 17x + 21, x]Principal subresultant coefficients are polynomials in the coefficients of input polynomials:
Subresultants[a x ^ 3 + b x ^ 2 + c x + d, x ^ 3 - 5b x - 7a, x]Options (3)
Modulus (3)
By default, the principal subresultant coefficients are computed over the rational numbers:
Subresultants[(x - 1) ^ 2(x - 2)(x - 3), (x - 1)(x - 4) ^ 2, x]Compute the principal subresultant coefficients over the integers modulo 2:
Subresultants[(x - 1) ^ 2(x - 2)(x - 3), (x - 1)(x - 4) ^ 2, x, Modulus -> 2]Compute the principal subresultant coefficients over the integers modulo 7:
Subresultants[(x - 1) ^ 2(x - 2)(x - 3), (x - 1)(x - 4) ^ 2, x, Modulus -> 7]Applications (2)
Find conditions for two polynomials to have exactly two common roots:
f = a x ^ 5 - (a - b + 1)x ^ 3 - b x + a;
g = 2x ^ 4 - 3x ^ 3 + 5x ^ 2 - 7x + 1;
psc = Subresultants[f, g, x];
cond = Reduce[psc[[1]] == 0 && psc[[2]] == 0 && psc[[3]] ≠ 0, {a, b}]Check that for the first solution f and g have exactly two common roots:
Reduce[(f //. First[{ToRules[cond]}]) == 0, x]//RootReduceReduce[g == 0, x]Find conditions for a quartic to have exactly two distinct roots:
f = a x ^ 4 + b x ^ 3 + c x ^ 2 + d x + e;
psc = Subresultants[f, D[f, x], x];
cond = Reduce[psc[[1]] == 0 && psc[[2]] == 0 && psc[[3]] ≠ 0, {a, b, c, d, e}]Check that for the first solution f has exactly two distinct roots:
Reduce[(f //. First[{ToRules[cond]}]) == 0, x]Properties & Relations (3)
Multiplicity of roots counts in determining the number of zero subresultants:
Subresultants[(x - 1)(x - 2)(x - 3), (x - 1)(x - 4)(x - 5), x]Subresultants[(x - 1) ^ 5(x - 2)(x - 3), (x - 1) ^ 4(x - 4)(x - 5), x]The length is determined by the minimum polynomial degree:
Length[Subresultants[x ^ 50 + a, x ^ 20 + b, x]]The first element of Subresultants is equal to Resultant:
f = 2x ^ 5 + 3x ^ 3 - 7x ^ 2 + 11x + 21;
g = -3x ^ 4 + 27x ^ 3 - 11x + 9;Subresultants[f, g, x]Resultant[f, g, x]Text
Wolfram Research (1999), Subresultants, Wolfram Language function, https://reference.wolfram.com/language/ref/Subresultants.html (updated 2022).
CMS
Wolfram Language. 1999. "Subresultants." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Subresultants.html.
APA
Wolfram Language. (1999). Subresultants. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Subresultants.html