Көпмүшенің түбірлерін анықтау үшін коэффициенттерінің ролі

Function of the coefficients of a polynomial that gives information on its roots

In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry. The discriminant of the quadratic polynomial is

the quantity which appears under the square root in the quadratic formula. If this discriminant is zero if and only if the polynomial has a double root. In the case of real coefficients, it is positive if the polynomial has two distinct real roots, and negative if it has two distinct complex conjugate roots. Similarly, the discriminant of a cubic polynomial is zero if and only if the polynomial has a multiple root. In the case of a cubic with real coefficients, the discriminant is positive if the polynomial has three distinct real roots, and negative if it has one real root and two distinct complex conjugate roots. More generally, the discriminant of a univariate polynomial of positive degree is zero if and only if the polynomial has a multiple root. For real coefficients and no multiple roots, the discriminant is positive if the number of non real roots is a multiple of 4 (including none), and negative otherwise. Several generalizations are also called discriminant: the discriminant of an algebraic number field; the discriminant of a quadratic form; and more generally, the discriminant of a form, of a homogeneous polynomial, or of a projective hypersurface (these three concepts are essentially equivalent).

The term "discriminant" was coined in 1851 by the British mathematician James Joseph Sylvester.

The discriminant of a linear polynomial (degree 1) is rarely considered. If needed, it is commonly defined to be equal to 1 (using the usual conventions for the empty product and considering that one of the two blocks of the Sylvester matrix is empty). There is no common convention for the discriminant of a constant polynomial (i. e., polynomial of degree 0). For small degrees, the discriminant is rather simple (see below), but for higher degrees, it may become unwieldy. For example, the discriminant of a general quartic has 16 terms, that of a quintic has 59 terms, and that of a sextic has 246 terms. This is OEIS sequence .

The discriminant of a polynomial over a field is zero if and only if the polynomial has a multiple root in some field extension. The discriminant of a polynomial over an integral domain is zero if and only if the polynomial and its derivative have a non constant common divisor. In characteristic 0, this is equivalent to saying that the polynomial is not square free (i. e., it is divisible by the square of a non constant polynomial). In nonzero characteristic p, the discriminant is zero if and only if the polynomial is not square free or it has an irreducible factor which is not separable (i. e., the irreducible factor is a polynomial in ).

In this section, all polynomials have real coefficients. It has been seen in that the sign of the discriminant provides useful information on the nature of the roots for polynomials of degree 2 and 3. For higher degrees, the information provided by the discriminant is less complete, but still useful. More precisely, for a polynomial of degree n, one has:
The polynomial has a multiple root if and only if its discriminant is zero. If the discriminant is positive, the number of non real roots is a multiple of 4. That is, there is a nonnegative integer k ≤ n/4 such that there are 2k pairs of complex conjugate roots and n − 4k real roots. If the discriminant is negative, the number of non real roots is not a multiple of 4. That is, there is a nonnegative integer k ≤ (n − 2)/4 such that there are 2k + 1 pairs of complex conjugate roots and n − 4k + 2 real roots.

The typical use of discriminants in algebraic geometry is for studying plane algebraic curves, and more generally algebraic hypersurfaces. Let V be such a curve or hypersurface; V is defined as the zero set of a multivariate polynomial. This polynomial may be considered as a univariate polynomial in one of the indeterminates, with polynomials in the other indeterminates as coefficients. The discriminant with respect to the selected indeterminate defines a hypersurface W in the space of the other indeterminates. The points of W are exactly the projection of the points of V (including the points at infinity), which either are singular or have a tangent hyperplane that is parallel to the axis of the selected indeterminate. For example, let f be a bivariate polynomial in X and Y with real coefficients, so that 1=f  = 0 is the implicit equation of a real plane algebraic curve. Viewing f as a univariate polynomial in Y with coefficients depending on X, then the discriminant is a polynomial in X whose roots are the X coordinates of the singular points, of the points with a tangent parallel to the Y axis and of some of the asymptotes parallel to the Y axis. In other words, the computation of the roots of the Y discriminant and the X discriminant allows one to compute all of the remarkable points of the curve, except the inflection points.

There are two classes of the concept of discriminant. The first class is the discriminant of an algebraic number field, which, in some cases including quadratic fields, is the discriminant of a polynomial defining the field. Discriminants of the second class arise for problems depending on coefficients, when degenerate instances or singularities of the problem are characterized by the vanishing of a single polynomial in the coefficients. This is the case for the discriminant of a polynomial, which is zero when two roots collapse. Most of the cases, where such a generalized discriminant is defined, are instances of the following. Let A be a homogeneous polynomial in n indeterminates over a field of characteristic 0, or of a prime characteristic that does not divide the degree of the polynomial. The polynomial A defines a projective hypersurface, which has singular points if and only the n partial derivatives of A have a nontrivial common zero. This is the case if and only if the multivariate resultant of these partial derivatives is zero, and this resultant may be considered as the discriminant of A. However, because of the integer coefficients resulting of the derivation, this multivariate resultant may be divisible by a power of n, and it is better to take, as a discriminant, the primitive part of the resultant, computed with generic coefficients. The restriction on the characteristic is needed because otherwise a common zero of the partial derivative is not necessarily a zero of the polynomial (see Euler's identity for homogeneous polynomials). In the case of a homogeneous bivariate polynomial of degree d, this general discriminant is times the discriminant defined in Several other classical types of discriminants, that are instances of the general definition are described in next sections.

The discriminant of an algebraic number field measures the size of the (ring of integers of the) algebraic number field. More specifically, it is proportional to the squared volume of the fundamental domain of the ring of integers, and it regulates which primes are ramified. The discriminant is one of the most basic invariants of a number field, and occurs in several important analytic formulas such as the functional equation of the Dedekind zeta function of K, and the analytic class number formula for K. A theorem of Hermite states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an open problem, and the subject of current research. Let K be an algebraic number field, and let OK be its ring of integers. Let b1, , bn be an integral basis of OK (i. e. a basis as a Z module), and let {σ1, , σn} be the set of embeddings of K into the complex numbers (i. e. injective ring homomorphisms K → C). The discriminant of K is the square of the determinant of the n by n matrix B whose (i,j) entry is σi(bj). Symbolically,

The discriminant of K can be referred to as the absolute discriminant of K to distinguish it from the of an extension K/L of number fields. The latter is an ideal in the ring of integers of L, and like the absolute discriminant it indicates which primes are ramified in K/L. It is a generalization of the absolute discriminant allowing for L to be bigger than Q; in fact, when L = Q, the relative discriminant of K/Q is the principal ideal of Z generated by the absolute discriminant of K.

Fundamental discriminants can also be characterized by their prime factorization. Consider the set consisting of prime numbers congruent to 1 modulo 4 (positive) and prime numbers congruent to 3 modulo 4 (negative):An integer is a fundamental discriminant if and only if it can be expressed as a product of pairwise relatively prime members of the set .