AskDefine | Define resultant

Dictionary Definition

resultant adj : following as an effect or result; "the period of tension and consequent need for military preparedness"; "the ensuant response to his appeal"; "the resultant savings were considerable"; "the health of the plants and the resulting flowers" [syn: consequent, ensuant, resulting(a), sequent]

Noun

1 the final point in a process [syn: end point]
2 something that results; "he listened for the results on the radio" [syn: result, final result, outcome, termination]
3 a vector that is the sum of two or more other vectors [syn: vector sum]

User Contributed Dictionary

English

Adjective

  1. following as a result or consequence of something

Noun

  1. anything that results from something else; an outcome
  2. a vector that is the vector sum of multiple vectors

Extensive Definition

In mathematics, the resultant of two monic polynomials P and Q over a field k is defined as the product
\mathrm(P,Q) = \prod_ (x-y),\,
of the differences of their roots, where x and y take on values in the algebraic closure of k. For non-monic polynomials with leading coefficients p and q, respectively, the above product is multiplied by
p^ q^.\,

Computation

  • When Q is separable, the above product can be rewritten to
\mathrm(P,Q) = \prod_ Q(x)\,
and this expression remains unchanged if Q is reduced modulo P. Note that, when non-monic, this includes the factor q^ but still needs the factor p^.
  • Let P' = P \mod Q. The above idea can be continued by swapping the roles of P' and Q. However, P' has a set of roots different from that of P. This can be resolved by writing \prod_ P'(y)\, as a determinant again, where P' has leading zero coefficients. This determinant can now be simplified by iterative expansion with respect to the column, where only the leading coefficient q of Q appears.
\mathrm(P,Q) = q^ \cdot \mathrm(P',Q)
Continuing this procedure ends up in a variant of the Euclidean algorithm. This procedure needs quadratic runtime.

Properties

  • \mathrm(P,Q) = (-1)^ \cdot \mathrm(Q,P)
  • \mathrm(P\cdot R,Q) = \mathrm(P,Q) \cdot \mathrm(R,Q)
  • If P' = P + R*Q and \deg P' = \deg P, then \mathrm(P,Q) = \mathrm(P',Q)
  • If X, Y, P, Q have the same degree and X = a_\cdot P + a_\cdot Q, Y = a_\cdot P + a_\cdot Q,
then \mathrm(X,Y) = \det^ \cdot \mathrm(P,Q)
  • \mathrm(P_-,Q) = \mathrm(Q_-,P) where P_-(z) = P(-z)

Applications

  • The resultant of a polynomial and its derivative is related to the discriminant.
  • Resultants can be used in algebraic geometry to determine intersections. For example, let
f(x,y)=0
and
g(x,y)=0
define algebraic curves in \mathbb^2_k. If f and g are viewed as polynomials in x with coefficients in k(y), then the resultant of f and g gives a polynomial in y whose roots are the y-coordinates of the intersection of the curves.

See also

References

resultant in German: Resultante
resultant in Spanish: Resultante
resultant in French: Résultant
resultant in Dutch: Resultante
resultant in Japanese: 終結式
resultant in Polish: Rugownik
resultant in Portuguese: Resultante

Synonyms, Antonyms and Related Words

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