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 .
- 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
- Stanley, Richard P. (1997). Enumerative Combinatorics, Volume 1. Cambridge University Press. ISBN 0-521-55309-1, 0-521-56069-1.
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