出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2016/04/27 04:08:14」(JST)
In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over their field of coefficients). In some older texts, the resultant is also called eliminant.^{[1]}
The resultant is widely used in number theory, either directly or through the discriminant, which is essentially the resultant of a polynomial and its derivative. The resultant of two polynomials with rational or polynomial coefficients may be computed efficiently on a computer. It is a basic tool of computer algebra, and is a builtin function of most computer algebra systems. It is used, among others, for cylindrical algebraic decomposition, integration of rational functions and drawing of curves defined by a bivariate polynomial equation.
The resultant of n homogeneous polynomials in n variables or multivariate resultant, sometimes called Macaulay's resultant, is a generalization of the usual resultant introduced by Macaulay. It is, with Gröbner bases, one of the main tools of effective elimination theory (elimination theory on computers).
For univariate monic polynomials P and Q over a field k, the resultant res(P,Q) is a polynomial function of their coefficients. It is defined as the product
of the differences of their roots in an algebraic closure of k; in the case of multiple roots, the factors are repeated according to their multiplicities. It results that the number of factors is always the product of the degrees of P and Q. For nonmonic polynomials with leading coefficients p and q, respectively, the above product is multiplied by p^{degQ}q^{degP}.
See the section on computation below, for a proof that res(P,Q) is a polynomial function of their coefficients.
Since the resultant depends polynomially (with integer coefficients) on the roots of P and Q, and it is invariant with respect to permutations of each set of roots, it must be possible to calculate it using an (integer) polynomial formula on the coefficients of P and Q. See elementary symmetric polynomial for details.
More concretely, the resultant is the determinant of the Sylvester matrix (and of the Bézout matrix) associated to P and Q. This is the standard definition of the resultant over a commutative ring.
The above definition of the resultant can be rewritten as
so it can be expressed polynomially in terms of the coefficients of Q for each fixed P. By the symmetry of the defining formula, the resultant is also a polynomial in the coefficients of P for each fixed Q. It follows that the resultant is a polynomial in the coefficients of P and Q jointly.
This expression remains unchanged if Q is replaced by the remainder P mod Q of the Euclidean division of Q by P. If we set P' = P mod Q, then this 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 res(P',Q) 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: res(P,Q)=q^{degPdegP'} res(P',Q). Continuing this procedure ends up in a variant of the Euclid's algorithm.
This procedure needs a number of arithmetic operations on the coefficients that is of the order of product of the degrees. However, when the coefficients are integers, rational numbers or polynomials, these arithmetic operations imply a number of GCD computations of coefficients which is of the same order and make the algorithm inefficient.
The subresultant pseudoremainder sequences were introduced to solve this problem and avoid any fraction and any GCD computation of coefficients. A more efficient algorithm is obtained by using the good behavior of the resultant under a ring homomorphism of the coefficients: to compute a resultant of two polynomials with integer coefficients, one computes their resultants modulo sufficiently many prime numbers and then reconstructs the result with the Chinese remainder theorem.
The resultant is sometimes defined for two homogeneous polynomials in two variables, in which case it vanishes when the polynomials have a common nonzero solution, or equivalently when they have a common zero on the projective line. More generally, the multipolynomial resultant,^{[2]} multivariate resultant or Macaulay's resultant of n homogeneous polynomials in n variables is a polynomial in their coefficients that vanishes when they have a common nonzero solution, or equivalently when the n hypersurfaces corresponding to them have a common zero in n–1 dimensional projective space. The multivariate resultant is, with Gröbner bases, one of the main tools of effective elimination theory (elimination theory on computers).

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リンク元  「結果」「sequence」「result」「outcome」「product」 