2024/05/27 by Binyamini, Gal · 1 citation
#03C64 #14D07 #14Q201 #58A17 #Algebraic Geometry (math.AG) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2405.16963
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 \mathbb RLN generated by LN-functions, as well as its expansion \mathbb RLN,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 \mathbb RLN,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 \mathbb Z-Hodge structures. We also give an effective Pila-Wilkie theorem for \mathbb RLN,exp-definable sets. Thus \mathbb RLN,exp can be used as an effective variant of \mathbb Ran,exp in the various applications of o-minimality to arithmetic geometry and Hodge theory.