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 q is d. Equivalently, a function is a quasi-polynomial if there exist polynomials such that when . The polynomials are called the constituents of f.
- Given a d-dimensional polytope P with rational vertices , define tP to be the convex hull of . The function is a quasi-polynomial in t of degree d. In this case, L(P,t) is a function . This is known as the Ehrhart quasi-polynomial, named after Eugène Ehrhart.
- Given two quasi-polynomials F and G, the convolution of F and G 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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