|
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. 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 built-in 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). ==Definition== 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 non-monic polynomials with leading coefficients p and q, respectively, the above product is multiplied by pdegQqdegP. See the section on computation below, for a proof that res(P,Q) is a polynomial function of their coefficients. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Resultant」の詳細全文を読む スポンサード リンク
|