2017/04/03 by Gal Binyamini, Binyamini, Gal
Computer Science · Mathematics · #03C64 #11G18 #34C10 #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1704.00442
openalex publication_date 2017/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Ω⊂\mathbb Rn be a relatively compact domain. A finite collection of real-valued functions on Ω is called a Noetherian chain if the partial derivatives of each function are expressible as polynomials in the functions. A Noetherian function is a polynomial combination of elements of a Noetherian chain. We introduce Noetherian parameters (degrees, size of the coefficients) which measure the complexity of a Noetherian chain. Our main result is an explicit form of the Pila-Wilkie theorem for sets defined using Noetherian equalities and inequalities: for any ε>0, the number of points of height H in the transcendental part of the set is at most C⋅ Hε where C can be explicitly estimated from the Noetherian parameters and ε. We show that many functions of interest in arithmetic geometry fall within the Noetherian class, including elliptic and abelian functions, modular functions and universal covers of compact Riemann surfaces, Jacobi theta functions, periods of algebraic integrals, and the uniformizing map of the Siegel modular variety Ag. We thus effectivize the (geometric side of) Pila-Zannier strategy for unlikely intersections in those instances that involve only compact domains.