A polynomial GCD certificate for exact flat bands in finite-range Bloch Hamiltonians

We formulate a polynomial GCD certificate for exact flat bands in finite-range periodic tight-binding Hamiltonians. Writing the characteristic polynomial of the Bloch Hamiltonian as a Laurent polynomial $P_L(\mathbf{z},\lambda)=\det(\lambda I-H_B(\mathbf{z}))=\sum_{\mathbf{t}}c_{\mathbf{t}}(\lambda)\mathbf{z}^{\mathbf{t}},$ we show that the monic greatest common divisor $G_L(\lambda)=\gcd_{\mathbf{t}}c_{\mathbf{t}}(\lambda)$ is precisely the maximum factor of $P_L$ that depends only on the energy variable. Its roots are exactly the exact flat-band energies, and their multiplicities give common algebraic multiplicities of these flat bands throughout the Brillouin zone. The coefficient-vanishing criterion underlying this statement is known in the flat-band and periodic-graph literature; the contribution emphasized here is the compact GCD formulation, its unit cell and Bloch-gauge invariance, and its use as a symbolic computation tool for hopping parameter engineering. The method is illustrated on kagome, dice and octahedron-chain examples, including weighted kagome and dice lattices. The certificate detects exact dispersionless eigenvalues; compact localized states, band touching and topological character must be analyzed in a subsequent eigenvector or projector calculation.