Periodicity in Quasipolynomial Convolution

The leading term of a convolution of quasipolynomials with periods p and q is periodic with period gcd(p, q), smaller than expected. The degree of the convolution is usually d+e+1; we characterize the exceptions. To do this we need to characterize the null space of a circulant matrix. We wish to point out a simple yet unexpected property of quasipolynomial calculus. A quasipolynomial is a function of positive integers that is given by a cyclically repeating sequence of polynomials; that is,