[Résolubilité d’une équation résultante et applications]
The large sieve is used to estimate the density of quadratic polynomials $Q\in \mathbb{Z}[x]$, such that there exists an odd degree polynomial defined over $\mathbb{Z}$ which has resultant $\pm 1$ with $Q$. Given a monic polynomial $R\in \mathbb{Z}[x]$ of odd degree, this is used to show that for almost all quadratic polynomials $Q\in \mathbb{Z}[x]$, there exists a prime $p$ such that $Q$ and $R$ share a common root in $\overline{\mathbb{F}}_p$. Using recent work of Landesman, an application to the average size of the odd part of the class group of quadratic number fields is also given.
Le grand crible est utilisé pour estimer la densité des polynômes quadratiques $Q \in \mathbb{Z}[x]$ tels qu’il existe un polynôme de degré impair défini sur $\mathbb{Z}$ dont le résultant avec $Q$ est égal à $\pm 1$. Étant donné un polynôme unitaire $R \in \mathbb{Z}[x]$ de degré impair, on s’en sert pour montrer que, pour presque tous les polynômes quadratiques $Q \in \mathbb{Z}[x]$, il existe un nombre premier $p$ tel que $Q$ et $R$ aient une racine commune dans $\overline{\mathbb{F}}_p$. En utilisant des travaux récents de Landesman, on obtient également une application concernant la taille moyenne de la partie impaire du groupe de classe des corps quadratiques.
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Keywords: Resultant, class number, large sieve
Mots-clés : Résultant, nombre de classes, grand crible
Tim Browning 1 ; Stephanie Chan 2
CC-BY 4.0
@article{JEP_2025__12__1677_0,
author = {Tim Browning and Stephanie Chan},
title = {Solubility of a resultant equation and applications},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1677--1691},
publisher = {\'Ecole polytechnique},
volume = {12},
year = {2025},
doi = {10.5802/jep.320},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.320/}
}
TY - JOUR AU - Tim Browning AU - Stephanie Chan TI - Solubility of a resultant equation and applications JO - Journal de l’École polytechnique — Mathématiques PY - 2025 SP - 1677 EP - 1691 VL - 12 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.320/ DO - 10.5802/jep.320 LA - en ID - JEP_2025__12__1677_0 ER -
%0 Journal Article %A Tim Browning %A Stephanie Chan %T Solubility of a resultant equation and applications %J Journal de l’École polytechnique — Mathématiques %D 2025 %P 1677-1691 %V 12 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.320/ %R 10.5802/jep.320 %G en %F JEP_2025__12__1677_0
Tim Browning; Stephanie Chan. Solubility of a resultant equation and applications. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1677-1691. doi: 10.5802/jep.320
[1] - “Paucity of rational points on fibrations with multiple fibres”, Algebra Number Theory 19 (2025) no. 10, p. 2049-2090 | DOI | MR | Zbl
[2] - Binary quadratic forms: Classical theory and modern computations, Springer-Verlag, New York, 1989 | DOI | MR | Zbl
[3] - “The 8-rank of the narrow class group and the negative Pell equation”, Forum Math. Sigma 10 (2022), article ID e46, 46 pages | DOI | MR | Zbl
[4] - “Heuristics on class groups of number fields”, in Number theory (Noordwijkerhout, 1983), Lect. Notes in Math., vol. 1068, Springer, Berlin, 1984, p. 33-62 | DOI | MR | Zbl
[5] - “On -torsion in class groups of number fields”, Algebra Number Theory 11 (2017) no. 8, p. 1739-1778 | DOI | MR
[6] - “On the negative Pell equation”, Ann. of Math. (2) 172 (2010) no. 3, p. 2035-2104 | DOI | MR | Zbl
[7] - “The parity of the period of the continued fraction of ”, Proc. London Math. Soc. (3) 101 (2010) no. 2, p. 337-391 | DOI | MR | Zbl
[8] - “Averages and higher moments for the -torsion in class groups”, Math. Ann. 379 (2021) no. 3-4, p. 1205-1229 | DOI | MR | Zbl
[9] - Opera de cribro, Amer. Math. Soc. Colloq. Publ., vol. 57, American Mathematical Society, Providence, RI, 2010 | DOI | MR | Zbl
[10] - “Densities for ranks of certain parts of -class groups”, Proc. Amer. Math. Soc. 99 (1987) no. 1, p. 1-8 | DOI | MR | Zbl
[11] - “Extension of conjectures of Cohen and Lenstra”, Exposition. Math. 5 (1987) no. 2, p. 181-184 | MR | Zbl
[12] - “Averages and moments associated to class numbers of imaginary quadratic fields”, Compositio Math. 153 (2017) no. 11, p. 2287-2309 | DOI | MR | Zbl
[13] - “On Stevenhagen’s conjecture”, 2022, to appear in Acta Math. | arXiv | Zbl
[14] - “Bounds for moments of -torsion in class groups”, Math. Ann. 390 (2024) no. 2, p. 3221-3237 | DOI | MR | Zbl
[15] - “A geometric approach to the Cohen-Lenstra heuristics”, J. Théor. Nombres Bordeaux 35 (2023) no. 3, p. 947-997 | DOI | Numdam | MR
[16] - “Inequalities for resultants and for decomposable forms”, in Diophantine approximation and its applications (Proc. Conf., Washington, D.C., 1972), Academic Press, New York-London, 1973, p. 235-253 | MR | Zbl
[17] - “Asymptotics for local solubility of diagonal quadrics over a split quadric surface”, 2024 | arXiv
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