Quasi-polynomial

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For quasi-polynomial time complexity of algorithms, see Quasi-polynomial time.

In mathematics, a quasi-polynomial (pseudo-polynomial) is a generalization of polynomials. While the coefficients of a polynomial come from a ring, the coefficients of quasi-polynomials are instead periodic functions with integral period. Quasi-polynomials appear throughout much of combinatorics as the enumerators for various objects.

A quasi-polynomial can be written as , where is a periodic function with integral period. If is not identically zero, then the degree of is . Equivalently, a function is a quasi-polynomial if there exist polynomials such that when . The polynomials are called the constituents of .

Examples[edit]

  • Given a -dimensional polytope with rational vertices , define to be the convex hull of . The function is a quasi-polynomial in of degree . In this case, is a function . This is known as the Ehrhart quasi-polynomial, named after Eugène Ehrhart.
  • Given two quasi-polynomials and , the convolution of and is

which is a quasi-polynomial with degree

See also[edit]

References[edit]