On Hua Loo-Keng's estimates of exponential sums in algebraic number fields

This paper provides a generalization of the Hua Loo-Keng estimation method of rational trigonometric sums with a polynomial in exponent in algebraic number fields, which are extensions of the field of rational numbers. In the ring of integers of this algebraic number field we consider integer and fractional ideals. For a complete system of residues for any integer ideal, Hua Loo-Keng proved an analogue of the Euler–Fourier formula, which, using results regarding the multiplicity of roots of a polynomial congruence modulo a prime ideal (“Hua Loo-Keng trees”), allows the problem to be reduced to the p-adic lifting of solutions, and this allows us to reduce the problem of estimating the sum to estimating the number of solutions of polynomial congruences modulo a power of a prime ideal. Furthermore, building upon Chen Jingrun's estimates in the field of rational numbers, we obtain improved constants for similar estimates in algebraic numeric fields.