| Title:
|
On the Győry-Sárközy-Stewart conjecture in function fields (English) |
| Author:
|
Shparlinski, Igor E. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
68 |
| Issue:
|
4 |
| Year:
|
2018 |
| Pages:
|
1067-1077 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
shifted polynomial product |
| Keyword:
|
number of zeros |
| MSC:
|
11R09 |
| MSC:
|
11S05 |
| MSC:
|
12E05 |
| idZBL:
|
Zbl 07031698 |
| idMR:
|
MR3881897 |
| DOI:
|
10.21136/CMJ.2018.0085-17 |
| . |
| Date available:
|
2018-12-07T06:21:49Z |
| Last updated:
|
2021-01-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147522 |
| . |
| Reference:
|
[1] Amoroso, F., Sombra, M., Zannier, U.: Unlikely intersections and multiple roots of sparse polynomials.Math. Z. 287 (2017), 1065-1081. Zbl 06819407, MR 3719528, 10.1007/s00209-017-1860-9 |
| Reference:
|
[2] Bernstein, D. J.: Sharper $ABC$-based bounds for congruent polynomials.J. Th{é}or. Nombres Bordx. 17 (2005), 721-725. Zbl 1093.11019, MR 2212120, 10.5802/jtnb.515 |
| Reference:
|
[3] Bombieri, E., Habegger, P., Masser, D., Zannier, U.: 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. Zbl 1209.11057, MR 2677603, 10.4171/RLM/570 |
| Reference:
|
[4] Bombieri, E., Masser, D., Zannier, U.: Intersecting a curve with algebraic subgroups of multiplicative groups.Int. Math. Res. Not. 20 (1999), 1119-1140. Zbl 0938.11031, MR 1728021, 10.1155/S1073792899000628 |
| Reference:
|
[5] Bombieri, E., Masser, D., Zannier, U.: On unlikely intersections of complex varieties with tori.Acta Arith. 133 (2008), 309-323. Zbl 1162.11031, MR 2457263, 10.4064/aa133-4-2 |
| Reference:
|
[6] Bugeaud, Y., Luca, F.: A quantitative lower bound for the greatest prime factor of $(ab+1)(bc+1)(ca+1)$.Acta Arith. 114 (2004), 275-294. Zbl 1122.11060, MR 2071083, 10.4064/aa114-3-3 |
| Reference:
|
[7] Corvaja, P., Zannier, U.: On the greatest prime factor of $(ab+1)(ac+1)$.Proc. Am. Math. Soc. 131 (2003), 1705-1709. Zbl 1077.11052, MR 1955256, 10.1090/S0002-9939-02-06771-0 |
| Reference:
|
[8] Corvaja, P., Zannier, U.: Some cases of Vojta's conjecture on integral points over function fields.J. Algebr. Geom. 17 (2008), 295-333. Zbl 1221.11146, MR 2369088, 10.1090/S1056-3911-07-00489-4 |
| Reference:
|
[9] Corvaja, P., Zannier, U.: An $abcd$ theorem over function fields and applications.Bull. Soc. Math. Fr. 139 (2011), 437-454. Zbl 1252.11031, MR 2869299, 10.24033/bsmf.2613 |
| Reference:
|
[10] Corvaja, P., Zannier, U.: 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. Zbl 1325.11060, MR 3082249, 10.4171/JEMS/409 |
| Reference:
|
[11] Győry, K., Sárközy, A.: On prime factors of integers of the form $(ab+1)(bc+1)(ca+1)$.Acta Arith. 79 (1997), 163-171. Zbl 0869.11071, MR 1438599, 10.4064/aa-79-2-163-171 |
| Reference:
|
[12] Győry, K., Sárközy, A., Stewart, C. L.: On the number of prime factors of integers of the form $ab+1$.Acta Arith. 74 (1996), 365-385. Zbl 0857.11047, MR 1378230, 10.4064/aa-74-4-365-385 |
| Reference:
|
[13] Habegger, P., Pila, J.: Some unlikely intersections beyond André-Oort.Compos. Math. 148 (2012), 1-27. Zbl 1288.11062, MR 2881307, 10.1112/S0010437X11005604 |
| Reference:
|
[14] Hernández, S., Luca, F.: On the largest prime factor of $(ab+1)(ac+1)(bc+1)$.Bol. Soc. Mat. Mex., III. Ser. 9 (2003), 235-244. Zbl 1108.11030, MR 2029272 |
| Reference:
|
[15] Mason, R. C.: Diophantine Equations Over Function Fields.London Mathematical Society Lecture Note Series 96, Cambridge University Press, Cambridge (1984). Zbl 0533.10012, MR 0754559, 10.1017/CBO9780511752490 |
| Reference:
|
[16] Maurin, G.: Équations multiplicatives sur les sous-variétés des tores.Int. Math. Res. Not. 2011 (2011), Article no. 23, 5259-5366 French. Zbl 1239.14020, MR 2855071, 10.1093/imrn/rnq248 |
| Reference:
|
[17] Ostafe, A.: On some extensions of the Ailon-Rudnick theorem.Monatsh. Math. 181 (2016), 451-471. Zbl 1355.11103, MR 3539944, 10.1007/s00605-016-0911-3 |
| Reference:
|
[18] Silverman, J. H.: The $S$-unit equation over function fields.Math. Proc. Camb. Philos. Soc. 95 (1984), 3-4. Zbl 0533.10013, MR 0727073, 10.1017/S0305004100061235 |
| Reference:
|
[19] Stewart, C. L., Tijdeman, R.: On the greatest prime factor of $(ab+1)(ac+1)(bc+1)$.Acta Arith. 79 (1997), 93-101. Zbl 0869.11072, MR 1438120, 10.4064/aa-79-1-93-101 |
| Reference:
|
[20] Stothers, W. W.: Polynomial identities and Hauptmoduln.Q. J. Math., Oxf. II. Ser. 32 (1981), 349-370. Zbl 0466.12011, MR 0625647, 10.1093/qmath/32.3.349 |
| Reference:
|
[21] Zannier, U.: Some problems of unlikely intersections in arithmetic and geometry.Annals of Mathematics Studies 181, Princeton University Press, Princeton (2012). Zbl 1246.14003, MR 2918151, 10.1515/9781400842711 |
| . |
