  In this paper, we consider the plasmon resonance in multi-layer structures.
The conductivity problem associated with uniformly distributed background field
is considered. We show that the plasmon mode is equivalent to the eigenvalue
problem of a matrix, whose order is the same to the number of layers. For any
number of layers, the exact characteristic polynomial is derived by a
conjecture and is verified by using induction. It is shown that all the roots
to the characteristic polynomial are real and exist in the span [-1, 2].
Numerical examples are presented for finding all the plasmon modes, and it is
surprisingly to find out that such multi-layer structures may induce so called
surface-plasmon-resonance-like band.
