Exact Order-Lifting Depth and Ramification in Global Function Fields

This preprint studies multiplicative order lifting in global function fields. For a rational function on a smooth projective curve over a finite field, the order-lifting depth at a unit place is shown to equal the local ramification index, without a separability assumption. For polynomials over finite fields, this gives an unrestricted valuation formula for Fermat quotients and explains the additional depth that can occur outside the large-degree range of Sauerberg and Shu. Hasse derivatives characterize every depth level and identify the kernels of higher quotient operators with the corresponding Hasse-vanishing spaces. The local ramification index also determines the multiplicative order modulo every power of the place and satisfies natural pullback and composition laws. For separable maps, logarithmic differentials and the Riemann–Hurwitz formula give global identities and bounds for the distribution of higher-depth places. The inseparable case reduces to the separable core after normalization. The results give a ramification-theoretic description of multiplicative order lifting and its relation to Fermat quotients and Hasse derivatives.