The Hilbert–Schinzel specialization property
-
Arnaud Bodin
,
Joachim König
Abstract
We establish a version “over the ring” of
the celebrated Hilbert Irreducibility Theorem. Given finitely many polynomials in
Funding source: Agence Nationale de la Recherche
Award Identifier / Grant number: ANR-11-LABX-0007-01
Award Identifier / Grant number: ANR-17-CE40–0023-01
Funding statement: The first and second authors were supported by the Labex CEMPI (ANR-11-LABX-0007-01). The first author was also supported by the ANR project “LISA” (ANR-17-CE40–0023-01), and thanks the University of British Columbia for his visit at PIMS.
References
[1] A. Bodin, P. Dèbes and S. Najib, Prime and coprime values of polynomials, Enseign. Math. 66 (2020), no. 1–2, 173–186. 10.4171/LEM/66-1/2-9Suche in Google Scholar
[2] A. Bodin, P. Dèbes and S. Najib, The Schinzel hypothesis for polynomials, Trans. Amer. Math. Soc. 373 (2020), no. 12, 8339–8364. 10.1090/tran/8198Suche in Google Scholar
[3] P. M. Cohn, Bezout rings and their subrings, Proc. Cambridge Philos. Soc. 64 (1968), 251–264. 10.1017/S0305004100042791Suche in Google Scholar
[4] T. Ekedahl, An infinite version of the Chinese remainder theorem, Comment. Math. Univ. St. Pauli 40 (1991), no. 1, 53–59. Suche in Google Scholar
[5] P. E. Frenkel and J. Pelikán, On the greatest common divisor of the value of two polynomials, Amer. Math. Monthly 124 (2017), no. 5, 446–450. 10.1142/S1793042118501518Suche in Google Scholar
[6] M. D. Fried and M. Jarden, Field arithmetic, 3rd ed., Ergeb. Math. Grenzgeb. (3) 11, Springer, Berlin 2008. Suche in Google Scholar
[7] S. Lang, Algebra, Addison-Wesley, Reading 1965. Suche in Google Scholar
[8] J. Neukirch, Algebraic number theory, Grundlehren Math. Wiss. 322, Springer, Berlin 1999. 10.1007/978-3-662-03983-0Suche in Google Scholar
[9] B. Poonen, Squarefree values of multivariable polynomials, Duke Math. J. 118 (2003), no. 2, 353–373. 10.1215/S0012-7094-03-11826-8Suche in Google Scholar
[10] A. Schinzel, A property of polynomials with an application to Siegel’s lemma, Monatsh. Math. 137 (2002), no. 3, 239–251. 10.1007/s00605-002-0491-2Suche in Google Scholar
[11] A. Schinzel and W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arith. 4 (1958), 185–208, erratum 5 (1958), 259. 10.4064/aa-4-3-185-208Suche in Google Scholar
[12] J.-P. Serre, Topics in Galois theory, Res. Notes Math. 1, Jones and Bartlett, Boston 1992. Suche in Google Scholar
[13] A. N. Skorobogatov and E. Sofos, Schinzel Hypothesis with probability 1 and rational points, preprint (2020), https://arxiv.org/abs/2005.02998. Suche in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- A central limit theorem for integrals of random waves
- Connectedness of Kisin varieties associated to absolutely irreducible Galois representations
- The Hilbert–Schinzel specialization property
- A mixed elliptic-parabolic boundary value problem coupling a harmonic-like map with a nonlinear spinor
- Isomorphisms among quantum Grothendieck rings and propagation of positivity
- The Neukirch–Uchida theorem with restricted ramification
- Perverse 𝔽p-sheaves on the affine Grassmannian
- Corrigendum to Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space (J. reine angew. Math. 727 (2017), 269–299)
Artikel in diesem Heft
- Frontmatter
- A central limit theorem for integrals of random waves
- Connectedness of Kisin varieties associated to absolutely irreducible Galois representations
- The Hilbert–Schinzel specialization property
- A mixed elliptic-parabolic boundary value problem coupling a harmonic-like map with a nonlinear spinor
- Isomorphisms among quantum Grothendieck rings and propagation of positivity
- The Neukirch–Uchida theorem with restricted ramification
- Perverse 𝔽p-sheaves on the affine Grassmannian
- Corrigendum to Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space (J. reine angew. Math. 727 (2017), 269–299)
