Gauss Sum of Index Four

Let p be a prime number, q=p^f and m|q-1. For a multiplicative character x of Fq with order m, the Gauss sum G(x) over Fq is defined

NONOSCILLATION AT A FINITE POINT FOR LINEAR HAMILTONIAN SYSTEM

Using the associated quadratic functional of the Hamiltonian system, we obtain the non-oscillation of all the prepared solutions for the Hamiltonian system at a finite point.