The Schinzel hypothesis for polynomials. (English) Zbl 1469.12003
The conjecture of Schinzel asserts that if \(P_1,P_2,\dots,P_k\in Z[x]\) are irreducible and their product does not have a fixed divisor \(>1\), then for infinitely \(n\) the values \(P_1(n),P_2(n),\dots,P_k(n)\) are prime [A. Schinzel and W. Sierpiński, Acta Arith. 4, 185–208 (1958; Zbl 0082.25802)]. The authors establish the truth of its analogue in the case when the ring \(Z\) of rational integers is replaced by the ring \(R_m=R[x_1,x_2,\dots,x_m]\), where \(m\ge1\) and \(R\) is a unique factorization domain with fraction field \(K\) in which the product formula holds and in case char\((K)=p>0\) one has \(K^p\ne K\) (Theorem 1.1). In Theorem 5.5 this result is extended to the case when the polynomials \(P_i\) are multivariate.
As a corollary one obtains the analogue of the Goldbach problem: if \(R\) satisfies the conditions of Theorem 1.1, then every non-constant polynomial in \(R_m\) is a sum of two irreducible polynomials, of which one is a binomial (Corollary 1.5). Earlier D. R. Hayes [Am. Math. Mon. 72, 45–46 (1965; Zbl 0128.04701)] did this in the case \(m=1, R=Z[x]\), pointing out that his proof works for principal ideal domains with infinitely many maximal ideals, and P. Pollack [Am. Math. Mon. 118, No. 1, 71–77 (2011; Zbl 1206.13010)] extended this (still for \(m=1\)) to the case when \(R\) is either a Noetherian domain having infinitely many maximal ideals or the ring of polynomials over an arbitrary integral domain.
As a corollary one obtains the analogue of the Goldbach problem: if \(R\) satisfies the conditions of Theorem 1.1, then every non-constant polynomial in \(R_m\) is a sum of two irreducible polynomials, of which one is a binomial (Corollary 1.5). Earlier D. R. Hayes [Am. Math. Mon. 72, 45–46 (1965; Zbl 0128.04701)] did this in the case \(m=1, R=Z[x]\), pointing out that his proof works for principal ideal domains with infinitely many maximal ideals, and P. Pollack [Am. Math. Mon. 118, No. 1, 71–77 (2011; Zbl 1206.13010)] extended this (still for \(m=1\)) to the case when \(R\) is either a Noetherian domain having infinitely many maximal ideals or the ring of polynomials over an arbitrary integral domain.
Reviewer: Władysław Narkiewicz (Wrocław)
MSC:
| 12E05 | Polynomials in general fields (irreducibility, etc.) |
| 11C08 | Polynomials in number theory |
| 12E25 | Hilbertian fields; Hilbert’s irreducibility theorem |
| 13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
References:
| [1] | Bodin, Arnaud; D\`ebes, Pierre; Najib, Salah, Irreducibility of hypersurfaces, Comm. Algebra, 37, 6, 1884-1900 (2009) · Zbl 1175.12001 · doi:10.1080/00927870802116562 |
| [2] | Bodin, Arnaud; D\`ebes, Pierre; Najib, Salah, Families of polynomials and their specializations, J. Number Theory, 170, 390-408 (2017) · Zbl 1380.12002 · doi:10.1016/j.jnt.2016.06.023 |
| [3] | Bodin, Arnaud, Reducibility of rational functions in several variables, Israel J. Math., 164, 333-347 (2008) · Zbl 1165.12001 · doi:10.1007/s11856-008-0033-2 |
| [4] | Bary-Soroker, Lior, Dirichlet’s theorem for polynomial rings, Proc. Amer. Math. Soc., 137, 1, 73-83 (2009) · Zbl 1259.12002 · doi:10.1090/S0002-9939-08-09474-4 |
| [5] | Bary-Soroker, Lior, Irreducible values of polynomials, Adv. Math., 229, 2, 854-874 (2012) · Zbl 1271.11114 · doi:10.1016/j.aim.2011.10.006 |
| [6] | Bender, Andreas O.; Wittenberg, Olivier, A potential analogue of Schinzel’s hypothesis for polynomials with coefficients in \({\mathbb{F}}_q[t]\), Int. Math. Res. Not., 36, 2237-2248 (2005) · Zbl 1087.11072 · doi:10.1155/IMRN.2005.2237 |
| [7] | Colliot-Th\'{e}l\`ene, Jean-Louis; Sansuc, Jean-Jacques, Sur le principe de Hasse et l’approximation faible, et sur une hypoth\`ese de Schinzel, Acta Arith., 41, 1, 33-53 (1982) · Zbl 0414.10009 · doi:10.4064/aa-41-1-33-53 |
| [8] | D\`ebes, Pierre, Density results for Hilbert subsets, Indian J. Pure Appl. Math., 30, 1, 109-127 (1999) · Zbl 0923.12001 |
| [9] | Michael D. Fried and Moshe Jarden, Field arithmetic, next edition. · Zbl 1055.12003 |
| [10] | Fried, Michael D.; Jarden, Moshe, Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 11, xxiv+792 pp. (2008), Springer-Verlag, Berlin · Zbl 1145.12001 |
| [11] | Harpaz, Yonatan; Wittenberg, Olivier, On the fibration method for zero-cycles and rational points, Ann. of Math. (2), 183, 1, 229-295 (2016) · Zbl 1342.14052 · doi:10.4007/annals.2016.183.1.5 |
| [12] | Kornblum, Heinrich; Landau, E., \"{U}ber die Primfunktionen in einer arithmetischen Progression, Math. Z., 5, 1-2, 100-111 (1919) · JFM 47.0154.02 · doi:10.1007/BF01203156 |
| [13] | Lang, Serge, Fundamentals of Diophantine geometry, xviii+370 pp. (1983), Springer-Verlag, New York · Zbl 0528.14013 · doi:10.1007/978-1-4757-1810-2 |
| [14] | Lang, Serge, Algebra, Graduate Texts in Mathematics 211, xvi+914 pp. (2002), Springer-Verlag, New York · Zbl 0984.00001 · doi:10.1007/978-1-4613-0041-0 |
| [15] | Najib, Salah, Sur le spectre d’un polyn\^ome \`a plusieurs variables, Acta Arith., 114, 2, 169-181 (2004) · Zbl 1071.12001 · doi:10.4064/aa114-2-6 |
| [16] | Najib, Salah, Une g\'{e}n\'{e}ralisation de l’in\'{e}galit\'{e} de Stein-Lorenzini, J. Algebra, 292, 2, 566-573 (2005) · Zbl 1119.13022 · doi:10.1016/j.jalgebra.2004.11.024 |
| [17] | Pollack, Paul, On polynomial rings with a Goldbach property, Amer. Math. Monthly, 118, 1, 71-77 (2011) · Zbl 1206.13010 · doi:10.4169/amer.math.monthly.118.01.071 |
| [18] | Rosen, Michael, Number theory in function fields, Graduate Texts in Mathematics 210, xii+358 pp. (2002), Springer-Verlag, New York · Zbl 1043.11079 · doi:10.1007/978-1-4757-6046-0 |
| [19] | Schinzel, Andrzej, Selected topics on polynomials, xxi+250 pp. (1982), University of Michigan Press, Ann Arbor, Mich. · Zbl 0487.12002 |
| [20] | Schinzel, A., Polynomials with special regard to reducibility, Encyclopedia of Mathematics and its Applications 77, x+558 pp. (2000), Cambridge University Press, Cambridge · Zbl 0956.12001 · doi:10.1017/CBO9780511542916 |
| [21] | Schinzel, A.; Sierpi\'{n}ski, W., Sur certaines hypoth\`eses concernant les nombres premiers, Acta Arith., 4, 185-208; erratum 5 (1958), 259 (1958) · Zbl 0082.25802 · doi:10.4064/aa-4-3-185-208 |
| [22] | Swan, Richard G., Factorization of polynomials over finite fields, Pacific J. Math., 12, 1099-1106 (1962) · Zbl 0113.01701 |
| [23] | Uchida, K\^oji, Separably Hilbertian fields, Kodai Math. J., 3, 1, 83-95 (1980) · Zbl 0435.12010 |
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