An algebraic method for the construction of the basic functionals of a Riemann surface, given in the form of a finite-sheeted covering of the sphere

Quantitative Riemann existence theorem over a number field

“…As we already indicated in the introduction, the prototype of our proof is the work of Zverovich [13]. Given a covering C x → P 1 and a point α ∈ P 1 , call the total ramification of x at α the quantity e(α) = e x (α) = (e 1 − 1) + · · · + (e s − 1), where e 1 , .…”

“…As we already indicated in the introduction, the prototype of our proof is the work of Zverovich [13]. Given a covering C where the unknown are the coefficients of variable polynomials F and Ψ, and, as in our argument, D(X) is the Y -discriminant of the variable polynomial F .…”

“…Using this, we estimate the height of the polynomial, and the degree and discriminant of number field generated by its coefficients. This argument is inspired by the work of Zverovich [13], who applies rather similar approach, though he works only in the complex domain. The system of equation considered by Zverovich is simpler than ours, but we could not understand one key point in his proof of the finiteness of the number of solutions.…”