Density of algebraic points on Noetherian varieties

Abstract

Let Ω ⊂ R n 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 · H ε 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.

Differential Equation Invariance Axiomatization

“…, h r }, with xed interpretations. 3 As a running example of such an extended term language, consider the unary function symbols exp, sin, cos 1:4 André Platzer and Yong Kiam Tan which are always interpreted as the real exponential and trigonometric functions respectively: e,ẽ ::= x | c | e +ẽ | e ·ẽ | exp(e) | sin(e) | cos(e) | (e) (1) Of course, the syntactic extension cannot be completely arbitrary, e.g., adding functions h that are nowhere di erentiable would fundamentally break the enterprise of studying ODEs directly by their local behavior. ese unsuitable syntactic extensions are ruled out by a set of extended term conditions.…”

“…e following de nition of Noetherian functions is standard, although the parameters that are used for studying the complexity of these functions [3,10,11] have been omi ed. e notation h :…”

“…e nal example highlights an important insight from Proposition 7.4. Even though this article only considers extended term languages with terms that are de ned everywhere, it is always possible to use logical formulas to implicitly characterize more terms, making use of closure properties of the Noetherian functions [3]. e following example illustrates implicit characterization of quotients which are de ned everywhere in the domain of interest: Example 7.11 (Implicit characterization of quotients).…”

“…is logical foundation is essential because it enables compositional syntactic reasoning for (continuous) di erential equations in isolation from other (discrete) parts of the system [30]. e class of Noetherian functions from real analytic geometry [3,10,11,43] meets all of the extended term conditions and thus provides an ideal se ing for extending term languages. is class includes many functions of practical interest for modeling hybrid systems, e.g., real exponential and trigonometric functions.…”

“…[3]). If h : H ⊆ R k → R is Noetherian and υ :ϒ ⊆ R l → R k has a compatible image υ(ϒ) ⊆ H where each component υ i : ϒ ⊆ R l → R for i = 1, .…”