Counting Theorems for Algebraic Relations
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866918450732990464 |
|---|---|
| author | Binyamini, Gal Hirata-Kohno, Noriko Kawashima, Makoto Salant, Yuval |
| author_facet | Binyamini, Gal Hirata-Kohno, Noriko Kawashima, Makoto Salant, Yuval |
| contents | Let X be a set definable in a sharply o-minimal structure. We consider the problem of counting the number of points where X intersects algebraic varieties V over Q of dimension k < codim X, as a function of T := deg(V) + h(V), where h(V) is the log-height of V. In particular, we conjecture that after removing a suitable "algebraic part", this number grows polynomially in T -- a generalization of Wilkie's conjecture. We show that this full conjecture implies some open problems in algebraic independence theory. We also formulate a weaker conjecture stating that all intersections above are contained in a poly(T) amount of balls of radius e^{-T}.
We then consider the case where X (subset of C^n) is a (compact piece of a) trajectory of a polynomial differential equation satisfying a variant of Nesterenko's D-property. Our main theorem is a proof of the weakened conjecture for such curves when k < sqrt(n) - 1. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15189 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counting Theorems for Algebraic Relations Binyamini, Gal Hirata-Kohno, Noriko Kawashima, Makoto Salant, Yuval Number Theory Algebraic Geometry Logic Let X be a set definable in a sharply o-minimal structure. We consider the problem of counting the number of points where X intersects algebraic varieties V over Q of dimension k < codim X, as a function of T := deg(V) + h(V), where h(V) is the log-height of V. In particular, we conjecture that after removing a suitable "algebraic part", this number grows polynomially in T -- a generalization of Wilkie's conjecture. We show that this full conjecture implies some open problems in algebraic independence theory. We also formulate a weaker conjecture stating that all intersections above are contained in a poly(T) amount of balls of radius e^{-T}. We then consider the case where X (subset of C^n) is a (compact piece of a) trajectory of a polynomial differential equation satisfying a variant of Nesterenko's D-property. Our main theorem is a proof of the weakened conjecture for such curves when k < sqrt(n) - 1. |
| title | Counting Theorems for Algebraic Relations |
| topic | Number Theory Algebraic Geometry Logic |
| url | https://arxiv.org/abs/2604.15189 |