Algebraic functions with infinitely many values in a number field
Fedor Pakovich1
1Department of Mathematics, Ben Gurion University of the Negev, Israel
Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 2, pp. 515-531

We provide several characterizations of algebraic curves $X : F(x, y) = 0 $ defined over $\overline{\mathbb{Q}}$ that satisfy the following property: there exist a number field $k$ and an infinite set $S \subset k$ such that, for every $y \in S$, the roots of the polynomial $F(x, y)$ belong to $k$.

Nous donnons plusieurs caractérisations des courbes algébriques $X : F(x, y) = 0 $ définies sur $\overline{\mathbb{Q}}$ qui satisfont la propriété suivante : il existe un corps de nombres $k$ et un ensemble infini $S \subset k$ tels que, pour tout $y \in S$, les racines du polynôme $F(x, y)$ appartiennent à $k$.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1370
Classification : 14H05, 37P15
Keywords: Algebraic Functions, Number Fields, Hilbert Irreducibility Theorem

Fedor Pakovich  1

1 Department of Mathematics, Ben Gurion University of the Negev, Israel
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Fedor Pakovich. Algebraic functions with infinitely many values in a number field. Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 2, pp. 515-531. doi: 10.5802/jtnb.1370
@article{JTNB_2026__38_2_515_0,
     author = {Fedor Pakovich},
     title = {Algebraic functions with infinitely many values in a number field},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {515--531},
     year = {2026},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {38},
     number = {2},
     doi = {10.5802/jtnb.1370},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1370/}
}
TY  - JOUR
AU  - Fedor Pakovich
TI  - Algebraic functions with infinitely many values in a number field
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2026
SP  - 515
EP  - 531
VL  - 38
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1370/
DO  - 10.5802/jtnb.1370
LA  - en
ID  - JTNB_2026__38_2_515_0
ER  - 
%0 Journal Article
%A Fedor Pakovich
%T Algebraic functions with infinitely many values in a number field
%J Journal de théorie des nombres de Bordeaux
%D 2026
%P 515-531
%V 38
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1370/
%R 10.5802/jtnb.1370
%G en
%F JTNB_2026__38_2_515_0

[1] Yuri Belotserkovskiĭ Effective analysis of a class of Diophantine equations, Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk, Volume 1988 (1988) no. 3, pp. 111-115 | Zbl

[2] Yuri Bilu Counting number fields in fibers (with an appendix by Jean Gillibert), Math. Z., Volume 288 (2018) no. 1-2, pp. 541-563 | Zbl | DOI | MR

[3] Yuri Bilu; Florian Luca Diversity in parametric families of number fields, Number theory – Diophantine problems, uniform distribution and applications, Springer, 2017, pp. 169-191 | Zbl | DOI

[4] Yuri Bilu; Florian Luca Number fields in fibers: the geometrically abelian case with rational critical values, Period. Math. Hung., Volume 75 (2017) no. 2, pp. 315-321 | Zbl | DOI | MR

[5] Peter Doyle; Curt McMullen Solving the quintic by iteration, Acta Math., Volume 163 (1989) no. 3-4, pp. 151-180 | Zbl | DOI | MR

[6] Roberto Dvornicich; Umberto Zannier Fields containing values of algebraic functions, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 21 (1994) no. 3, pp. 421-443 | Zbl | Numdam | MR

[7] Roberto Dvornicich; Umberto Zannier Fields containing values of algebraic functions. II: On a conjecture of Schinzel, Acta Arith., Volume 73 (1995) no. 3, pp. 201-210 | Zbl | DOI | MR

[8] Gerd Faltings Complements to Mordell, Rational points. Seminar Bonn/Wuppertal 1983/84 (Aspects of Mathematics), Volume E6, Vieweg & Sohn, 1984, pp. 203-227 | DOI | MR

[9] Hershel M. Farkas; Irwin Kra Riemann surfaces, Graduate Texts in Mathematics, 71, Springer, 1992, xvi+363 pages | Zbl | DOI | MR

[10] Michael D. Fried Introduction to modular towers: generalizing dihedral group-modular curve connections, Recent developments in the inverse Galois problem (Contemporary Mathematics), Volume 186, American Mathematical Society, 1995, pp. 111-171 | Zbl | DOI

[11] Ernesto Girondo; Gabino González-Diez Introduction to compact Riemann surfaces and dessins d’enfants, London Mathematical Society Student Texts, 79, London Mathematical Society, 2012, xii+298 pages | Zbl | MR

[12] John Milnor On Lattès maps, Dynamics on the Riemann Sphere, European Mathematical Society, 2006, pp. 9-43 | Zbl | DOI

[13] Fedor Pakovich On semiconjugate rational functions, Geom. Funct. Anal., Volume 26 (2016) no. 4, pp. 1217-1243 | Zbl | DOI | MR

[14] Fedor Pakovich On algebraic curves ${A}(x)-{B}(y)=0$ of genus zero, Math. Z., Volume 288 (2018) no. 1-2, pp. 299-310 | Zbl | DOI | MR

[15] Fedor Pakovich On rational functions whose normalization has genus zero or one, Acta Arith., Volume 182 (2018) no. 1, pp. 73-100 | Zbl | DOI | MR

[16] Fedor Pakovich Semiconjugate rational functions: a dynamical approach, Arnold Math. J., Volume 4 (2018) no. 1, pp. 59-68 | Zbl | DOI | MR

[17] Fedor Pakovich Recomposing rational functions, Int. Math. Res. Not., Volume 2019 (2019) no. 7, pp. 1921-1935 | Zbl | DOI | MR

[18] Fedor Pakovich On generalized Lattès maps, J. Anal. Math., Volume 142 (2020) no. 1, pp. 1-39 | Zbl | DOI | MR

[19] Fedor Pakovich Tame rational functions: Decompositions of iterates and orbit intersections, J. Eur. Math. Soc., Volume 25 (2023) no. 10, pp. 3953-3978 | Zbl | DOI | MR

[20] Fedor Pakovich Lower bounds for genera of fiber products, Isr. J. Math., Volume 269 (2025) no. 1, pp. 475-500 | Zbl | DOI | MR

[21] Umberto Zannier An effective solution of a certain Diophantine problem, Rend. Semin. Mat. Univ. Padova, Volume 93 (1995), pp. 177-183 | Zbl

Cité par Sources :