Nonpolynomial

Nonpolynomial refers to a function or expression that is not a polynomial in a given variable. A polynomial in x over a field is a finite sum a0 + a1 x + … + an x^n, where the coefficients ai are from the field and n is a nonnegative integer. The term nonpolynomial thus captures any function that cannot be written in that finite polynomial form.

Nonpolynomial functions can be further categorized. Some are algebraic without being polynomials, meaning they satisfy a polynomial equation relating the dependent and independent variables. For example, y = sqrt(x) satisfies y^2 = x, so it is algebraic but not a polynomial in x. Others are transcendental, meaning they do not satisfy any polynomial equation with constant coefficients; common examples include exp(x), sin(x), and ln(x) (the latter on its natural domain). The distinction between algebraic and transcendental is a stronger classification than merely “nonpolynomial,” but both fall under the broad umbrella of nonpolynomial expressions.

Domain considerations are important. Nonpolynomials may be defined only on restricted domains (such as ln(x) requiring x > 0 or complex-valued functions with branch cuts). They can also exhibit behaviors that polynomials do not, such as infinite series expansions, oscillations, or rapid growth.

In mathematics, identifying whether a function is polynomial or nonpolynomial helps determine its property class, possible representations, and analytic behavior. The term serves as a general descriptor for expressions that cannot be captured by a finite polynomial formula in the variable of interest. Examples include exp(x), sin(x), sqrt(x), |x|, and x^pi.