42 research outputs found
On the degree of convergence of lemniscates in finite connected domains
AbstractFor an arbitrary bounded closed set E in the complex plane with complement Ω of finite connectivity, we study the degree of convergence of the lemniscates in Ω
A minimum principle for plurisubharmonic functions
The main goal of this work is to give new and precise generalizations to
various classes of plurisubharmonic functions of the classical minimum modulus
principle for holomorphic functions of one complex variable, in the spirit of
the famous lemma of Cartan-Boutroux. As an application we obtain precise
estimates on the size of "plurisubharmonic lemniscates" in terms of appropriate
Hausdorff contents
Analytic curves in algebraic varieties over number fields
We establish algebraicity criteria for formal germs of curves in algebraic
varieties over number fields and apply them to derive a rationality criterion
for formal germs of functions, which extends the classical rationality theorems
of Borel-Dwork and P\'olya-Bertrandias valid over the projective line to
arbitrary algebraic curves over a number field.
The formulation and the proof of these criteria involve some basic notions in
Arakelov geometry, combined with complex and rigid analytic geometry (notably,
potential theory over complex and -adic curves). We also discuss geometric
analogues, pertaining to the algebraic geometry of projective surfaces, of
these arithmetic criteria.Comment: 55 pages. To appear in "Algebra, Arithmetic, and Geometry: In Honor
of Y.i. Manin", Y. Tschinkel & Yu. Manin editors, Birkh\"auser, 200