Funktionsgrade

Funktionsgrade is a mathematical term used to describe the degree of a function, most often in the context of polynomial functions. In German mathematical literature the expression Funktionsgrad or Grad der Funktion appears, though in many cases the simple word Grad is used when referring to polynomials.

In the univariate case, if f is a polynomial f(x) = a_n x^n + … + a_1 x + a_0 with a_n ≠ 0, then der Funktionsgrad of f is n, the highest exponent with a nonzero coefficient. For polynomials in several variables, f(x1, …, xm) = ∑ aα x^α, where α is a multi-index, the Funktionsgrad is the maximum total degree max|α| among all terms with aα ≠ 0, and |α| = α1 + … + αm.

In the context of rational functions, degrees can be assigned to the numerator and denominator separately (deg(numerator) and deg(denominator)). The overall degree of the function is not always a single defined number; in some usages people refer to the difference deg(numerator) − deg(denominator) or describe the degrees of the components.

Limitations and scope: For non-polynomial functions (such as exponential, logarithmic, or trigonometric functions), the term Funktionsgrad is not standard and is generally not defined in the same way as for polynomials. In approximation theory, one may speak of the order of a Taylor polynomial or the degree of an approximation rather than a Funktionsgrad of the function itself.

See also: Degree (mathematics), Polynomial degree, Multivariate polynomial, Polynomial function.