Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 68, No. 4, pp. 1067-1077, 2018

On the Győry-Sárközy-Stewart conjecture in function fields

Igor E. Shparlinski

Received February 28, 2017. Published online April 20, 2018.

Abstract: We consider function field analogues of the conjecture of Győry, Sárközy and Stewart (1996) on the greatest prime divisor of the product $(ab+1)(ac+1)(bc+1)$ for distinct positive integers $a$, $b$ and $c$. In particular, we show that, under some natural conditions on rational functions $F,G,H \in{\mathbb C}(X)$, the number of distinct zeros and poles of the shifted products $FH+1$ and $GH+1$ grows linearly with $\deg H$ if $\deg H \ge\max\{\deg F, \deg G\} $. We also obtain a version of this result for rational functions over a finite field.

Keywords: shifted polynomial product; number of zeros

Classification MSC: 11R09, 11S05, 12E05

DOI: 10.21136/CMJ.2018.0085-17

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

References:
[1] F. Amoroso, M. Sombra, U. Zannier: Unlikely intersections and multiple roots of sparse polynomials. Math. Z. 287 (2017), 1065-1081. DOI 10.1007/s00209-017-1860-9 | MR 3719528 | Zbl 06819407
[2] D. J. Bernstein: Sharper $ABC$-based bounds for congruent polynomials. J. Théor. Nombres Bordx. 17 (2005), 721-725. DOI 10.5802/jtnb.515 | MR 2212120 | Zbl 1093.11019
[3] E. Bombieri, P. Habegger, D. Masser, U. Zannier: A note on Maurin's theorem. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 21 (2010), 251-260. DOI 10.4171/RLM/570 | MR 2677603 | Zbl 1209.11057
[4] E. Bombieri, D. Masser, U. Zannier: Intersecting a curve with algebraic subgroups of multiplicative groups. Int. Math. Res. Not. 20 (1999), 1119-1140. DOI 10.1155/S1073792899000628 | MR 1728021 | Zbl 0938.11031
[5] E. Bombieri, D. Masser, U. Zannier: On unlikely intersections of complex varieties with tori. Acta Arith. 133 (2008), 309-323. DOI 10.4064/aa133-4-2 | MR 2457263 | Zbl 1162.11031
[6] Y. Bugeaud, F. Luca: A quantitative lower bound for the greatest prime factor of $(ab+1)(bc+1)(ca+1)$. Acta Arith. 114 (2004), 275-294. DOI 10.4064/aa114-3-3 | MR 2071083 | Zbl 1122.11060
[7] P. Corvaja, U. Zannier: On the greatest prime factor of $(ab+1)(ac+1)$. Proc. Am. Math. Soc. 131 (2003), 1705-1709. DOI 10.1090/S0002-9939-02-06771-0 | MR 1955256 | Zbl 1077.11052
[8] P. Corvaja, U. Zannier: Some cases of Vojta's conjecture on integral points over function fields. J. Algebr. Geom. 17 (2008), 295-333. DOI 10.1090/S1056-3911-07-00489-4 | MR 2369088 | Zbl 1221.11146
[9] P. Corvaja, U. Zannier: An $abcd$ theorem over function fields and applications. Bull. Soc. Math. Fr. 139 (2011), 437-454. DOI 10.24033/bsmf.2613 | MR 2869299 | Zbl 1252.11031
[10] P. Corvaja, U. Zannier: Greatest common divisors of $u-1$, $v-1$ in positive characteristic and rational points on curves over finite fields. J. Eur. Math. Soc. (JEMS) 15 (2013), 1927-1942. DOI 10.4171/JEMS/409 | MR 3082249 | Zbl 1325.11060
[11] K. Győry, A. Sárközy: On prime factors of integers of the form $(ab+1)(bc+1)(ca+1)$. Acta Arith. 79 (1997), 163-171. DOI 10.4064/aa-79-2-163-171 | MR 1438599 | Zbl 0869.11071
[12] K. Győry, A. Sárközy, C. L. Stewart: On the number of prime factors of integers of the form $ab+1$. Acta Arith. 74 (1996), 365-385. DOI 10.4064/aa-74-4-365-385 | MR 1378230 | Zbl 0857.11047
[13] P. Habegger, J. Pila: Some unlikely intersections beyond André-Oort. Compos. Math. 148 (2012), 1-27. DOI 10.1112/S0010437X11005604 | MR 2881307 | Zbl 1288.11062
[14] S. Hernández, F. Luca: On the largest prime factor of $(ab+1)(ac+1)(bc+1)$. Bol. Soc. Mat. Mex., III. Ser. 9 (2003), 235-244. MR 2029272 | Zbl 1108.11030
[15] R. C. Mason: Diophantine Equations Over Function Fields. London Mathematical Society Lecture Note Series 96, Cambridge University Press, Cambridge (1984). DOI 10.1017/CBO9780511752490 | MR 0754559 | Zbl 0533.10012
[16] G. Maurin: Équations multiplicatives sur les sous-variétés des tores. Int. Math. Res. Not. 2011 (2011), Article no. 23, 5259-5366. (In French.) DOI 10.1093/imrn/rnq248 | MR 2855071 | Zbl 1239.14020
[17] A. Ostafe: On some extensions of the Ailon-Rudnick theorem. Monatsh. Math. 181 (2016), 451-471. DOI 10.1007/s00605-016-0911-3 | MR 3539944 | Zbl 1355.11103
[18] J. H. Silverman: The $S$-unit equation over function fields. Math. Proc. Camb. Philos. Soc. 95 (1984), 3-4. DOI 10.1017/S0305004100061235 | MR 0727073 | Zbl 0533.10013
[19] C. L. Stewart, R. Tijdeman: On the greatest prime factor of $(ab+1)(ac+1)(bc+1)$. Acta Arith. 79 (1997), 93-101. DOI 10.4064/aa-79-1-93-101 | MR 1438120 | Zbl 0869.11072
[20] W. W. Stothers: Polynomial identities and Hauptmoduln. Q. J. Math., Oxf. II. Ser. 32 (1981), 349-370. DOI 10.1093/qmath/32.3.349 | MR 0625647 | Zbl 0466.12011
[21] U. Zannier: Some problems of unlikely intersections in arithmetic and geometry. Annals of Mathematics Studies 181, Princeton University Press, Princeton (2012). DOI 10.1515/9781400842711 | MR 2918151 | Zbl 1246.14003

Affiliations: Igor E. Shparlinski, Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia, e-mail: igor.shparlinski@unsw.edu.au

Springer subscribers can access the papers on Springer website.

[List of online first articles] [Contents of Czechoslovak Mathematical Journal] [Full text of the older issues of Czechoslovak Mathematical Journal at DML-CZ]

PDF available at:

Springer
for subscribers

···

Institute of Mathematics CAS
free access

···

Digital Mathematics Library
free access