Log-Noetherian functions

We introduce the class of Log-Noetherian (LN) functions. These
are holomorphic solutions to algebraic differential equations (in several variables) with logarithmic singularities. We prove an upper bound on the number of solutions for systems of LN equations, resolving in particular Khovanskii’s conjecture for Noetherian functions. Consequently, we show that the structure R_LN generated by LN-functions, as well as its expansion R_LN,exp, are effectively o-minimal: definable sets in these structures admit effective bounds on their complexity in terms of the complexity of the defining formulas.
We show that R_LN,exp contains the horizontal sections of regular flat connections with quasiunipotent monodromy over algebraic varieties. It therefore
contains the universal covers of Shimura varieties and period maps of polarized
variations of Z-Hodge structures. We also give an effective Pila-Wilkie theorem
for R_LN,exp-definable sets. Thus R_LN,exp can be used as an effective variant
of R_an,exp in the various applications of o-minimality to arithmetic geometry
and Hodge theory.