The Davenport-Lewis-Schinzel problem on the reducibility of $f(X)-g(Y)$
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914430091001856 |
|---|---|
| author | Behajaina, Angelot König, Joachim Neftin, Danny |
| author_facet | Behajaina, Angelot König, Joachim Neftin, Danny |
| contents | We solve the problem of Davenport--Lewis--Schinzel (DLS), originating in the 1950s, regarding the reducibility of $f(X)-g(Y)\in\mathbb C[X,Y]$. This yields an almost-complete solution to the Hilbert--Siegel problem: For a polynomial map $f$ whose composition factors avoid only very specific low-degree polynomials, we explicitly describe over which integers the fibers of $f$ are reducible. We further apply the solution to stability of iterates of $f$ in arithmetic dynamics, and to solving the functional equation $f(X)=g(Y)$ in $X,Y\in\mathbb{C}(z)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27728 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Davenport-Lewis-Schinzel problem on the reducibility of $f(X)-g(Y)$ Behajaina, Angelot König, Joachim Neftin, Danny Number Theory Algebraic Geometry Group Theory We solve the problem of Davenport--Lewis--Schinzel (DLS), originating in the 1950s, regarding the reducibility of $f(X)-g(Y)\in\mathbb C[X,Y]$. This yields an almost-complete solution to the Hilbert--Siegel problem: For a polynomial map $f$ whose composition factors avoid only very specific low-degree polynomials, we explicitly describe over which integers the fibers of $f$ are reducible. We further apply the solution to stability of iterates of $f$ in arithmetic dynamics, and to solving the functional equation $f(X)=g(Y)$ in $X,Y\in\mathbb{C}(z)$. |
| title | The Davenport-Lewis-Schinzel problem on the reducibility of $f(X)-g(Y)$ |
| topic | Number Theory Algebraic Geometry Group Theory |
| url | https://arxiv.org/abs/2603.27728 |