Counting Special Points: Logic, Diophantine Geometry, and Transcendence Theory
نویسنده
چکیده
We expose a theorem of Pila and Wilkie on counting rational points in sets definable in o-minimal structures and some applications of this theorem to problems in diophantine geometry due to Masser, Peterzil, Pila, Starchenko, and Zannier.
منابع مشابه
Open Diophantine Problems
Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel’s Conjecture). Some questions related to Mahler’s ...
متن کامل7 Elliptic Functions and Transcendence
Transcendental numbers form a fascinating subject: so little is known about the nature of analytic constants that more research is needed in this area. Even when one is interested only in numbers like π and eπ that are related to the classical exponential function, it turns out that elliptic functions are required (so far, this should not last forever!) to prove transcendence results and get a ...
متن کاملElliptic Functions and Transcendence
Transcendental numbers form a fascinating subject: so little is known about the nature of analytic constants that more research is needed in this area. Even when one is interested only in numbers like π and e which are related with the classical exponential function, it turns out that elliptic functions are required (so far – this should not last for ever!) to prove transcendence results and ge...
متن کاملRational Points
is known to have only finitely many triples of positive integer solutions x, y, z for a given n > 2 (Faltings, 1983). In Chapter 11, special situations are described in which more precise information is accessible. For example, if x is in S, then n is bounded by a computable number C5 = Cb(pv ..., p8). From these examples, it should be clear that the book is a mine of information for workers in...
متن کاملA Proof of the André - Oort Conjecture via Mathematical Logic
INTRODUCTION Extending work of Bombieri and Pila on counting lattice points on convex curves [3], Pila and Wilkie proved a strong counting theorem on the number of rational points in a more general class of sets definable in an o-minimal structure on the real numbers [37]. Following a strategy proposed by Zannier, the Pila-Wilkie upper bound has been leveraged against Galois-theoretic lower bou...
متن کامل