Quasi-polynomial

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. 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 q ( k ) = c d ( k ) k d + c d − 1 ( k ) k d − 1 + ⋯ + c 0 ( k ) {displaystyle q(k)=c_{d}(k)k^{d}+c_{d-1}(k)k^{d-1}+cdots +c_{0}(k)} , where c i ( k ) {displaystyle c_{i}(k)} is a periodic function with integral period. If c d ( k ) {displaystyle c_{d}(k)} is not identically zero, then the degree of q {displaystyle q} is d {displaystyle d} . Equivalently, a function f : N → N {displaystyle fcolon mathbb {N} o mathbb {N} } is a quasi-polynomial if there exist polynomials p 0 , … , p s − 1 {displaystyle p_{0},dots ,p_{s-1}} such that f ( n ) = p i ( n ) {displaystyle f(n)=p_{i}(n)} when n ≡ i mod s {displaystyle nequiv i{mod {s}}} . The polynomials p i {displaystyle p_{i}} are called the constituents of f {displaystyle f} . which is a quasi-polynomial with degree ≤ deg ⁡ F + deg ⁡ G + 1. {displaystyle leq deg F+deg G+1.}