Geodesic
Improvements on dimension growth results and effective Hilbert’s irreducibility theorem
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e153

Voir la notice de l'article provenant de la source Cambridge University Press

We sharpen and generalize the dimension growth bounds for the number of points of bounded height lying on an irreducible algebraic variety of degree d, over any global field. In particular, we focus on the affine hypersurface situation by relaxing the condition on the top degree homogeneous part of the polynomial describing the affine hypersurface, while sharpening the dependence on the degree in the bounds compared to previous results. We formulate a conjecture about plane curves which provides a conjectural approach to the uniform degree $3$ case (the only remaining open case). For induction on dimension, we develop a higher-dimensional effective version of Hilbert’s irreducibility theorem, which is of independent interest. 
Cluckers, Raf; Dèbes, Pierre; Hendel, Yotam I.; Nguyen, Kien Huu; Vermeulen, Floris. Improvements on dimension growth results and effective Hilbert’s irreducibility theorem. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e153. doi: 10.1017/fms.2025.10096
@article{10_1017_fms_2025_10096,
     author = {Cluckers, Raf and D\`ebes, Pierre and Hendel, Yotam I. and Nguyen, Kien Huu and Vermeulen, Floris},
     title = {Improvements on dimension growth results and effective {Hilbert{\textquoteright}s} irreducibility theorem},
     journal = {Forum of Mathematics, Sigma},
     pages = {e153},
     year = {2025},
     volume = {13},
     number = {1},
     doi = {10.1017/fms.2025.10096},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2025.10096/}
}
TY  - JOUR
AU  - Cluckers, Raf
AU  - Dèbes, Pierre
AU  - Hendel, Yotam I.
AU  - Nguyen, Kien Huu
AU  - Vermeulen, Floris
TI  - Improvements on dimension growth results and effective Hilbert’s irreducibility theorem
JO  - Forum of Mathematics, Sigma
PY  - 2025
SP  - e153
VL  - 13
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fms.2025.10096/
DO  - 10.1017/fms.2025.10096
ID  - 10_1017_fms_2025_10096
ER  - 
%0 Journal Article
%A Cluckers, Raf
%A Dèbes, Pierre
%A Hendel, Yotam I.
%A Nguyen, Kien Huu
%A Vermeulen, Floris
%T Improvements on dimension growth results and effective Hilbert’s irreducibility theorem
%J Forum of Mathematics, Sigma
%D 2025
%P e153
%V 13
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/fms.2025.10096/
%R 10.1017/fms.2025.10096
%F 10_1017_fms_2025_10096

[BCK25] Binyamini, G., Cluckers, R. and Kato, F., ‘Sharp bounds for the number of rational points on algebraic curves and dimension growth, over all global fields’, Proc. Lond. Math. Soc. 130(1) (2025), e70016.10.1112/plms.70016 Google Scholar | DOI

[BCN24] Binyamini, G., Cluckers, R. and Novikov, D., ‘Bounds for rational points on algebraic curves, optimal in the degree, and dimension growth’, Int. Math. Res. Not. IMRN (2024), rnae034. Google Scholar

[BHB05] Browning, T. D. and Heath-Brown, D. R., ‘Counting rational points on hypersurfaces’, J. Reine Angew. Math. 584 (2005), 83–115.10.1515/crll.2005.2005.584.83 Google Scholar | DOI

[BHB17] Browning, T. D. and Heath-Brown, D. R., ‘Forms in many variables and differing degrees’, J. Eur. Math. Soc. (JEMS) 19(2) (2017), 357–394. Google Scholar | DOI

[BHBS06] Browning, T. D., Heath-Brown, D. R. and Salberger, P., ‘Counting rational points on algebraic varieties’, Duke Math. J. 132(3) (2006), 545–578. Google Scholar

[BP89] Bombieri, E. and Pila, J., ‘The number of integral points on arcs and ovals’, Duke Math. J. 59(2) (1989), 337–357.10.1215/S0012-7094-89-05915-2 Google Scholar | DOI

[Bro09] Browning, T. D., Quantitative Arithmetic of Projective Varieties, Progress in Mathematics, vol. 277 (Birkhäuser, Basel, 2009). Google Scholar

[BS07] Boissiere, S. and Sarti, A., ‘Counting lines on surfaces’, Ann. Sc. Norm. Super. Pisa Cl. Sci. 6(1) (2007), 39–52. Google Scholar

[BST+20] Bhargava, M., Shankar, A., Taniguchi, T., Thorne, F., Tsimerman, J. and Zhao, Y., ‘Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves’, J. Amer. Math. Soc. 33(4) (2020), 1087–1099.10.1090/jams/945 Google Scholar | DOI

[CCDN20] Castryck, W., Cluckers, R., Dittmann, Ph. and Nguyen, K. H., ‘The dimension growth conjecture, polynomial in the degree and without logarithmic factors’, Algebra Number Theory 14(8) (2020), 2261–2294.10.2140/ant.2020.14.2261 Google Scholar | DOI

[CFL20] Cluckers, R., Forey, A. and Loeser, F., ‘Uniform Yomdin–Gromov parametrizations and points of bounded height in valued fields’, Algebra Number Theory 14(6) (2020), 1423–1456.10.2140/ant.2020.14.1423 Google Scholar | DOI

[DGH21] Dimitrov, V., Gao, Z. and Habegger, P., ‘Uniformity in Mordell–Lang for curves’, Ann. Math. (2) 194(1) (2021), 237–298. Google Scholar

[DW08] Dèbes, P. and Walkowiak, Y., ‘Bounds for Hilbert’s irreducibility theorem’, Pure Appl. Math. Q. 4(4) (2008), Special Issue: In honor of Jean-Pierre Serre. Part 1, 1059–1083.10.4310/PAMQ.2008.v4.n4.a4 Google Scholar | DOI

[FJ23] Fried, M. D. and Jarden, M., Field Arithmetic, 4th corrected ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 11 (Springer, Cham, 2023).10.1007/978-3-031-28020-7 Google Scholar | DOI

[Har77] Hartshorne, R., Algebraic Geometry (Springer Science+Business Media, 1977).10.1007/978-1-4757-3849-0 Google Scholar | DOI

[Har95] Harris, J., Algebraic Geometry, Graduate Texts in Mathematics, vol. 133 (Springer-Verlag, New York, 1995), corrected reprint of the 1992 original. Google Scholar

[HB83] Heath-Brown, D. R., ‘Cubic forms in ten variables’, Proc. Lond. Math. Soc. (3) 47(2) (1983), 225–257. Google Scholar | DOI

[HB02] Heath-Brown, D. R., ‘The density of rational points on curves and surfaces’, Ann. Math. (2) 155(2) (2002), 553–595.10.2307/3062125 Google Scholar | DOI

[HS00] Hindry, M. and Silverman, J.H., Diophantine Geometry (Springer, New York, 2000). Google Scholar | DOI

[Kal95] Kaltofen, E., ‘Effective Noether irreducibility forms and applications’, J. Comput. System Sci. 50(2) (1995), 274–295. 23rd Symposium on the Theory of Computing (New Orleans, LA, 1991). Google Scholar

[Lan83] Lang, S., Fundamentals of Diophantine Geometry (Springer, New York, 1983).10.1007/978-1-4757-1810-2 Google Scholar | DOI

[Pil95] Pila, J., ‘Density of integral and rational points on varieties’, Astérisque (1995), no. 228, 183–187. Columbia University Number Theory Seminar (New York, 1992). Google Scholar

[PS22] Paredes, M. and Sasyk, R., ‘Uniform bounds for the number of rational points on varieties over global fields’, Algebra Number Theory 16(8) (2022), 1941–2000.10.2140/ant.2022.16.1941 Google Scholar | DOI

[PS24] Paredes, M. and Sasyk, R., ‘Effective Hilbert’s irreducibility theorem for global fields’, Israel J. Math. 261(2) (2024), 851–877.10.1007/s11856-023-2604-7 Google Scholar | DOI

[Rup86] Ruppert, W., ‘Reduzibilität ebener Kurven’, J. Reine Angew. Math. 369 (1986), 167–191. Google Scholar

[Ryd03] Rydh, D., Chow Varieties, Master’s thesis, Royal Institute of Technology, Stockholm, June 2003. Available at: https://people.kth.se/~dary/Chow.pdf Google Scholar

[Sal15] Salberger, P., ‘Uniform bounds for rational points on cubic hypersurfaces’, in Arithmetic and Geometry, London Math. Soc. Lecture Note Ser., vol. 420 (Cambridge Univ. Press, Cambridge, 2015), 401–421.10.1017/CBO9781316106877.021 Google Scholar | DOI

[Sal23a] Salberger, P., ‘Counting integral points of affine hypersurfaces’, preprint (2023), . Google Scholar | arXiv

[Sal23b] Salberger, P., ‘Counting rational points on projective varieties’, Proc. Lond. Math. Soc. (3) 126(4) (2023), 1092–1133.10.1112/plms.12508 Google Scholar | DOI

[Sch69] Schinzel, A., ‘An improvement of Runge’s theorem on Diophantine equations’, Comment. Pontificia Acad. Sci. 2(20) (1969), 1–9. Google Scholar

[Sch85] Schmidt, W. M., ‘Integer points on curves and surfaces’, Monatsh. Math. 99(1) (1985), 45–72.10.1007/BF01300739 Google Scholar | DOI

[Sed17] Sedunova, A., ‘On the Bombieri–Pila method over function fields’, Acta Arith. 181(4) (2017), 321–331.10.4064/aa8613-8-2017 Google Scholar | DOI

[Seg43] Segre, B., ‘The maximum number of lines lying on a quartic surface’, Q. J. Math. 14(1) (1943), 86–96.10.1093/qmath/os-14.1.86 Google Scholar | DOI

[Ser89] Serre, J.-P., Lectures on the Mordell–Weil Theorem, Aspects of Mathematics, E15 (Friedr. Vieweg & Sohn, Braunschweig, 1989). Translated from the French and edited by Brown, Martin from notes by Waldschmidt, Michel. Google Scholar | DOI

[Ser92] Serre, J.-P., Topics in Galois Theory, Research Notes in Mathematics, vol. 1 (Jones and Bartlett, Boston, MA, 1992). Lecture notes prepared by Henri Darmon, with a foreword by Darmon and the author. Google Scholar

[Spr98] Springer, T. A., Linear Algebraic Groups, Progress in Mathematics, vol. 9 (Birkhäuser, 1998). Google Scholar | DOI

[SZ95] Schinzel, A. and Zannier, U., ‘The least admissible value of the parameter in Hilbert’s irreducibility theorem’, Acta Arith. 69(3) (1995), 293–302. Google Scholar

[Ver22] Vermeulen, F., ‘Points of bounded height on curves and the dimension growth conjecture over ’, Bull. Lond. Math. Soc. 54(2) (2022), 635–654. Google Scholar | DOI

[Ver24] Vermeulen, F., ‘Dimension growth for affine varieties’, Int. Math. Res. Not. IMRN 2024(15) (2024), 11464–11483.10.1093/imrn/rnae137 Google Scholar | DOI

[Wal05] Walkowiak, Y., ‘Théorème d’irréductibilité de Hilbert effectif’, Acta Arith. 116(4) (2005), 343–362.10.4064/aa116-4-3 Google Scholar | DOI

[Wal15] Walsh, M. N., ‘Bounded rational points on curves’, Int. Math. Res. Not. IMRN 2015(14) (2015), 5644–5658. Google Scholar

[Zha05] Zhang, S.-W., ‘Equidistribution of CM-points on quaternion Shimura varieties’, Int. Math. Res. Not. 2005(59) (2005), 3657–3689. Google Scholar

Cité par Sources :