Quasipolynomials

Quasipolynomials are functions that, on the integers, behave like polynomials with periodically varying coefficients. More precisely, a function f is a quasipolynomial of degree at most d if there exists a positive integer m and functions a_0(n), a_1(n), ..., a_d(n) that are periodic with period m such that f(n) = a_d(n) n^d + a_{d-1}(n) n^{d-1} + ... + a_0(n) for all integers n. Equivalently, f can be expressed as a linear combination of binomial coefficients with coefficients that are periodic functions. When all the coefficient functions are constant (period 1), f is a genuine polynomial.

Quasipolynomials arise naturally in combinatorics and number theory, notably in lattice-point counting. A central example is in Ehrhart theory: for a rational convex polytope P, the function L_P(t) that counts lattice points in the dilate tP is a quasipolynomial in t of degree equal to the dimension of P. The period divides the common denominator of P’s vertex coordinates; if P is an integral polytope, L_P(t) is actually a polynomial.

Key properties include the degree and leading coefficient, which are preserved in the sense that the leading coefficient is constant, while lower-degree coefficients may be periodic. The period of a sum of quasipolynomials is the least common multiple of the individual periods, and the product has period dividing the product of periods.

Examples include floor(n^2/2), which is a degree-2 quasipolynomial with period 2, and lattice-point counting functions for dilated rational polygons. Quasipolynomials thus generalize polynomials to accommodate systematic periodic fluctuations in discrete counting problems.