  We study the structure of the digit sets ${\mathcal D}$ for the integral
self-similar tiles $T(b,{\mathcal{D}})$ (we call such ${\mathcal D}$ a {\it
tile digit set} with respect to $b$). So far the only available classes of such
tile digit sets are the complete residue sets and the product-forms. Our
investigation here is based on the spectrum of the mask polynomial $P_{\mathcal
D}$, i.e., the zeros of $P_{\mathcal D}$ on the unit circle. By using the
Fourier criteria of self-similar tiles of Kenyon and Protasov, as well as the
algebraic techniques of cyclotomic polynomial, we characterize the tile digit
sets through some product of cyclotomic polynomials (kernel polynomials), which
is a generalization of the product-form to higher order.
