polynomial

polynomial A formal power series, i.e. a sum of multiples of powers of an independent variable known as the indeterminate (often written as x, s, or t), e.g. 3x⁴ + 7x² + 2x + 5

or, in general,

The coefficients (a_i) are elements of some algebraic system, S, having appropriate addition and multiplication operations; the expression is then described as a polynomial over S. For example, if the coefficients are all integers, the polynomial is said to be over the integers. If a_r ≠ 0 but a_i = 0 for all i > r, then r is called the degree of the polynomial, usually written r = deg (p)

If a_r = 1, the polynomial is monic.

Arithmetic on polynomials consists primarily of addition, subtraction, and multiplication of polynomials; in some cases division, factoring, and taking the greatest common divisor are also important operations.

Addition and subtraction are done by adding or subtracting the coefficients of like powers of x.

Multiplication is done by the rule (a_rx^r + … a₁x + a₀)(b_sx^s + … b₁x + b₀) = (c_r+sx^r+s + … c₁x + c₀)

where c_k = a₀b_k + a₁b_k–1 + … a_k–1b₁ + a_kb₀ a_i,b_j = 0 for i > r, j > s

In coding theory, much use is made of polynomials over the ring of integers modulo q, for some integer q > 1. Such polynomials themselves form a commutative ring with an identity. More particularly, coding theory employs polynomials over the field of integers modulo p, for some suitable prime number p. (For binary systems, p = 2.) These polynomials can be multiplied and divided; in general, they may be factorized. A polynomial (over a field) that can be factorized is said to be reducible; otherwise it is irreducible. When divided by another, a polynomial over a field gives a unique quotient and remainder. Every such polynomial can be uniquely factorized into irreducible factors.

The set of polynomials (over a field), modulo a given monic irreducible polynomial (over the same field), itself forms a field; this is called an extension field of the original base field of coefficients (which were integers modulo p). Extension fields of this kind are fundamental to much of coding theory.

The extension field of polynomials modulo G, over the integers modulo p, contains p^g elements, where g is the degree of G. G is called the generating polynomial of the extension field. A polynomial that is an element of this field is said to be primitive if and only if it does not exactly divide the polynomial x^c – 1 (over the field of integers modulo p) for any c less than p^g – 1.

A practical problem of some importance is to find all the values of x that satisfy the equation p_n(x) = 0

where p_n(x) is a polynomial equation of degree n. Such equations have n solutions, called roots, which in general are complex. If the given coefficients a_i are real the complex roots occur in conjugate pairs. It is quite common for some of the roots to be very sensitive to small changes in the coefficients, i.e. to have a large condition number.

A single root α may be found by an iteration such as Newton's method or the secant method. The polynomial p_n–1(x) = p_n(x)/(x – α)

has the same roots as p_n except for α; it may be used to determine the other roots. The process of calculating p_n–1 is known as deflation, and is used after each root is found; thus the polynomials used are of progressively lower degree. Deflation depends on the roots being accurate. If an approximate root is used, the deflated polynomial will have inaccurate coefficients, and possibly very inaccurate roots. To minimize deterioration of the successive polynomials used, it is important to determine each root to the greatest possible precision and, where feasible, to determine the roots in increasing order of magnitude.