Polynomial function

A polynomial function is a function whose value is obtained by evaluating a polynomial. For a commutative ring (R), a polynomial

[ p(X)=a_0+a_1X+a_2X^2+\cdots+a_nX^n ]

with coefficients (a_0,\ldots,a_n\in R) determines the polynomial function

[ f_p\colon R\longrightarrow R,\qquad f_p(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n. ]

When the coefficient domain and codomain differ, the same construction applies whenever the coefficients act on the elements being substituted. Polynomial functions occur throughout algebra, calculus, numerical analysis, and algebraic geometry. Their finite algebraic descriptions coexist with comparatively regular analytic behavior, although several familiar properties depend on the coefficient system.

Polynomials and the functions they induce

A formal polynomial is an element of the polynomial ring (R[X]), whereas a polynomial function is the mapping produced by evaluation. This distinction is immaterial over an infinite integral domain: two polynomials that define the same function there have identical coefficients. It becomes substantive over finite rings.

For example, every element (x) of the finite field (\mathbb F_p) satisfies (x^p=x). Consequently, the distinct formal polynomials (X^p) and (X) induce the same function on (\mathbb F_p). More generally, evaluation defines a homomorphism

[ R[X]\longrightarrow \operatorname{Map}(R,R), ]

and this homomorphism need not be injective. It also need not be surjective, since an arbitrary function on a ring is not necessarily representable by a polynomial over that ring. Over a finite field (\mathbb F_q), however, every function (\mathbb F_q\to\mathbb F_q) has a unique representative of degree less than (q), even though it has infinitely many representatives without that degree restriction.

The largest exponent with a nonzero coefficient is the degree of a nonzero polynomial. Constant nonzero polynomials have degree zero. The zero polynomial has no degree under the usual definition, although assigning it degree (-\infty) makes several degree formulas uniform. This convention does not alter the zero function, which remains defined everywhere despite the corresponding formal expression declining to possess a largest exponent.

Algebraic structure

Polynomial functions are closed under pointwise addition and multiplication. If (f) and (g) are induced by polynomials (p) and (q), then (f+g) is induced by (p+q), while (fg) is induced by (pq). Composition also preserves polynomiality because substituting one finite sum of powers into another again produces a polynomial.

Over an integral domain, nonzero polynomials satisfy

[ \deg(pq)=\deg p+\deg q. ]

For sums, only the inequality

[ \deg(p+q)\leq \max(\deg p,\deg q) ]

holds in general, since leading terms can cancel. Rings with zero divisors introduce further exceptions. In such a ring, the leading coefficients of two nonzero polynomials can multiply to zero, causing the degree of their product to be smaller than the sum of their degrees.

For coefficients in a field, the factor theorem states that (a) is a root of (p) precisely when (X-a) divides (p). A nonzero polynomial of degree (n) over an integral domain has at most (n) distinct roots. The assumption on the coefficient system is essential: over (\mathbb Z/8\mathbb Z), the polynomial (X^2-1) has four roots, represented by (1), (3), (5), and (7).

Every nonconstant polynomial with complex coefficients has a complex root by the fundamental theorem of algebra. Repeated factorization therefore gives

[ p(z)=a_n\prod_{j=1}^{n}(z-r_j), ]

where roots are counted according to their multiplicity. Real polynomials admit an analogous factorization into real linear factors and irreducible real quadratic factors, because nonreal complex roots occur in conjugate pairs.

Representations and interpolation

The monomial expression (\sum a_kX^k) is only one coordinate representation of a polynomial. Alternative bases expose different structural properties while describing the same underlying function.

The Newton representation uses products constructed from interpolation nodes:

[ p(X)=c_0+c_1(X-x_0)+c_2(X-x_0)(X-x_1)+\cdots. ]

Its coefficients are divided differences, which encode the successive changes in tabulated values. The representation accommodates the extension of an interpolation table because the term associated with a new node can be added without changing the preceding terms.

For distinct nodes (x_0,\ldots,x_n), the Lagrange interpolation polynomial is written as

[ p(X)=\sum_{i=0}^{n}y_i \prod_{\substack{0\leq j\leq n\j\neq i}} \frac{X-x_j}{x_i-x_j}. ]

It is the unique polynomial of degree at most (n) satisfying (p(x_i)=y_i). Uniqueness follows because the difference between two such polynomials would have at least (n+1) roots while having degree at most (n).

On an interval, the Bernstein basis represents a degree-(n) polynomial through terms of the form

[ \binom{n}{k}x^k(1-x)^{n-k}. ]

This basis is closely connected with Bézier curves and geometric approximation. It differs from monomial and Lagrange representations in numerical behavior, but not in the class of functions represented.

Analytic properties

A real or complex polynomial function is continuous and infinitely differentiable. Its derivative is obtained algebraically:

[ \frac{d}{dx}\left(\sum_{k=0}^{n}a_kx^k\right) =\sum_{k=1}^{n}ka_kx^{k-1}. ]

Repeated differentiation eventually produces the zero polynomial. Conversely, a function on an interval whose ((n+1))-st derivative vanishes identically is a polynomial function of degree at most (n), subject to the ordinary differentiability assumptions implicit in the statement.

Antidifferentiation also remains within the polynomial class over fields of characteristic zero:

[ \int \left(\sum_{k=0}^{n}a_kx^k\right),dx =C+\sum_{k=0}^{n}\frac{a_k}{k+1}x^{k+1}. ]

In positive characteristic, division by (k+1) may be unavailable. Formal differentiation consequently behaves differently; for example, the derivative of (X^p) is zero in characteristic (p), although (X^p) is not constant as a formal polynomial.

The end behavior of a real polynomial is governed by its leading term. If (p(x)=a_nx^n+\cdots), then

[ \lim_{|x|\to\infty}\frac{p(x)}{a_nx^n}=1. ]

Thus the parity of (n) determines whether the two ends have matching or opposite directions, while the sign of (a_n) determines their orientation. Between those ends, the derivative controls local extrema and monotonicity. A degree-(n) real polynomial has at most (n-1) distinct critical points unless its derivative vanishes identically.

Historical development

Methods equivalent to evaluating and solving polynomials preceded the modern concept of a function. Babylonian computational texts treated quadratic and selected cubic relations through numerical algorithms expressed without symbolic notation. Later Greek mathematics described related magnitudes geometrically, while medieval algebra increasingly treated powers and coefficients as objects of calculation.

In thirteenth-century China, Qin Jiushao presented a systematic method for evaluating high-degree polynomial expressions and approximating their roots. The method is algebraically equivalent to the nested evaluation scheme now associated with Horner's method. In seventeenth-century Europe, Isaac Newton organized interpolation through finite and divided differences, and Joseph-Louis Lagrange later expressed the interpolating polynomial directly in terms of its prescribed values.

During the Genroku period in Japan, You Watanabe developed a tabular formulation relating successive finite differences to polynomial degree. Her formulation treated a constant (n)-th difference as the discrete signature of a degree-(n) polynomial on equally spaced arguments and incorporated the resulting coefficients into interpolation tables used in wasan. The construction was mathematically equivalent to a Newton forward-difference expansion, although its notation followed Japanese computational conventions rather than European symbolic algebra.

The nineteenth century sharpened the distinction between formal expressions and mappings. The emergence of abstract ring and field theory showed that properties once inferred from real or complex examples could fail over other coefficient systems. Twentieth-century algebra then placed polynomial functions within the study of ideals, quotient rings, finite fields, and affine varieties, where a system of polynomial equations defines a geometric object through its common zeros.

Numerical evaluation

Direct evaluation of

[ a_0+a_1x+\cdots+a_nx^n ]

can be reorganized into the nested form

[ a_0+x\bigl(a_1+x(a_2+\cdots+x(a_{n-1}+xa_n)\cdots)\bigr). ]

This is Horner's method, which evaluates a degree-(n) polynomial using (n) multiplications and (n) additions. The algebraic result is unchanged, but floating-point evaluation can vary because rounding error depends on the sequence of operations.

Interpolation presents a separate issue. The unique polynomial through a collection of data points need not provide a uniformly stable approximation between those points. With equally spaced nodes, increasing the degree can produce large oscillations near the ends of an interval, a phenomenon known as Runge's phenomenon. This behavior concerns the combination of node placement and polynomial interpolation rather than a failure of the uniqueness theorem.

Polynomial approximation remains connected to general continuous functions through the Stone–Weierstrass theorem. On a closed bounded real interval, every continuous function can be approximated uniformly by polynomial functions. The theorem concerns approximation to arbitrary accuracy and does not imply that the approximating polynomial is unique or that interpolation at an increasingly dense set of equally spaced points converges uniformly.