The theory of the transformation of homogeneous polynomials, or forms (see ALGEBRAIC FORMS) by linear substitutions in the variables, and of the invariants and covariants associated with such transformation, is an important branch of modern algebra, with numerous applications. For example, if x, y in the poly nomial are expressed, in terms of a new pair of variables u, v, by the relations where ab-13-y= I, it is found that is identically equal to an expression of the form A in which — 4A C = 4o,c , a which is of fundamental significance in analytic geometry. More generally, for any values of a, j3, y, 6, 4 C = (a6 — 4ac) • The expression itself a polynomial in terms of the co efficients a, b, c, is called an invariant.
Polynomials in one variable are the simplest class of functions from the point of view of the calculus, because the rules for their differentiation and integration are particularly simple, and are obtained immediately from the definitions of these processes. The result of differentiating or integrating a polynomial with respect to its independent variable is always a polynomial.
In the modern theory of functions (see FUNCTION), any poly nomial is a continuous and analytic function of its variables. If a function of a single complex variable z is analytic for every finite value of z, and becomes infinite when z, represented by a point in a plane, goes to infinity in an arbitrary manner, the func tion is necessarily a polynomial.
One of the chief investigators of the properties of polynomials during the 19th century was the Russian mathematician Chebichev (Tschebyschef) (1821-94). Among theorems discovered only recently may be mentioned (a) the one which states that, if a polynomial of the nth degree in x does not exceed a number L in absolute value for values of x in the interval from —1 to +I, the absolute value of its derivative can not exceed in the same interval (S. Bernstein, 1912), and (b) some results on the relation between the roots of a polynomial and those of its derivative in the complex plane (J. L. Walsh, Annals of Mathe matics, 1920, and subsequent papers in the Bulletin and Trans actions of the American Mathematical Society).
Applications.—Apart from their specific properties, poly nomials are of fundamental importance from their use in the approximate representation of other functions. The standard func tions of elementary analysis can be represented by power series (see SERIES), of the form or, more generally, which reduces to the preceding when a= o; the sum of an infinite series is by definition the limit approached by the sum of a finite number of its terms, as the number of terms is taken larger and larger, and the sum of a finite number of terms of a power series is a polynomial. Representation by power series can be made the
basis for a systematic treatment of analytic functions of a com plex variable. Another important form of development in series, theoretically applicable with greater generality, proceeds in terms of the polynomials of Legendre (1752-1833) or Legendre's co efficients. These may be defined as the coefficients of successive powers of r in the power series for (1 One of their most striking properties is that the product of any two of them, integrated over the interval from to +1, gives zero. The theory of Legendre series is still under investigation. Approxima tions in terms of the polynomials of Hermite (1822-1901) are of importance in the theory of probability. Weierstrass (1885) proved that an arbitrary continuous function can be uniformly approximated by a polynomial with any assigned degree of accuracy.