  Coherence, the superposition of orthogonal quantum states, is indispensable
in various quantum processes. Inspired by the polynomial invariant for
classifying and quantifying entanglement, we first define polynomial coherence
measure and systematically investigate its properties. Except for the qubit
case, we show that there is no polynomial coherence measure satisfying the
criterion that its value takes zero if and only if for incoherent states. Then,
we release this strict criterion and obtain a necessary condition for
polynomial coherence measure. Furthermore, we give a typical example of
polynomial coherence measure for pure states and extend it to mixed states via
a convex-roof construction. Analytical formula of our convex-roof polynomial
coherence measure is obtained for symmetric states which are invariant under
arbitrary basis permutation. Consequently, for general mixed states, we give a
lower bound of our coherence measure.
