# Effective bounds for the zeros of linear recurrences in function fields

Clemens Fuchs[1]; Attila Pethő[2]

• [1] Institut für Mathematik Technische Universität Graz Steyrergasse 30 8010 Graz, Austria Current Address: Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, Postbus 9512, 2300 RA Leiden, The Netherlands
• [2] Institute of Informatics University of Debrecen, Debrecen Pf. 12 4010 Debrecen, Hungary
• Volume: 17, Issue: 3, page 749-766
• ISSN: 1246-7405

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## Abstract

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In this paper, we use the generalisation of Mason’s inequality due to Brownawell and Masser (cf. [8]) to prove effective upper bounds for the zeros of a linear recurring sequence defined over a field of functions in one variable.Moreover, we study similar problems in this context as the equation ${G}_{n}\left(x\right)={G}_{m}\left(P\left(x\right)\right),\left(m,n\right)\in {ℕ}^{2}$, where $\left({G}_{n}\left(x\right)\right)$ is a linear recurring sequence of polynomials and $P\left(x\right)$ is a fixed polynomial. This problem was studied earlier in [14,15,16,17,32].

## How to cite

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Fuchs, Clemens, and Pethő, Attila. "Effective bounds for the zeros of linear recurrences in function fields." Journal de Théorie des Nombres de Bordeaux 17.3 (2005): 749-766. <http://eudml.org/doc/249449>.

@article{Fuchs2005,
abstract = {In this paper, we use the generalisation of Mason’s inequality due to Brownawell and Masser (cf. [8]) to prove effective upper bounds for the zeros of a linear recurring sequence defined over a field of functions in one variable.Moreover, we study similar problems in this context as the equation $G_n(x)=G_m(P(x)), (m,n)\in \{\mathbb\{N\}\}^2$, where $(G_n(x))$ is a linear recurring sequence of polynomials and $P(x)$ is a fixed polynomial. This problem was studied earlier in [14,15,16,17,32].},
affiliation = {Institut für Mathematik Technische Universität Graz Steyrergasse 30 8010 Graz, Austria Current Address: Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, Postbus 9512, 2300 RA Leiden, The Netherlands; Institute of Informatics University of Debrecen, Debrecen Pf. 12 4010 Debrecen, Hungary},
author = {Fuchs, Clemens, Pethő, Attila},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {3},
pages = {749-766},
publisher = {Université Bordeaux 1},
title = {Effective bounds for the zeros of linear recurrences in function fields},
url = {http://eudml.org/doc/249449},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Fuchs, Clemens
AU - Pethő, Attila
TI - Effective bounds for the zeros of linear recurrences in function fields
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 3
SP - 749
EP - 766
AB - In this paper, we use the generalisation of Mason’s inequality due to Brownawell and Masser (cf. [8]) to prove effective upper bounds for the zeros of a linear recurring sequence defined over a field of functions in one variable.Moreover, we study similar problems in this context as the equation $G_n(x)=G_m(P(x)), (m,n)\in {\mathbb{N}}^2$, where $(G_n(x))$ is a linear recurring sequence of polynomials and $P(x)$ is a fixed polynomial. This problem was studied earlier in [14,15,16,17,32].
LA - eng
UR - http://eudml.org/doc/249449
ER -

## References

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1. A. Baker, New advances in transcendence theory. Cambridge Univ. Press, Cambridge, 1988. Zbl0644.00005MR971989
2. F. Beukers, The multiplicity of binary recurrences. Compositio Math. 40 (1980), 251–267. Zbl0396.10005MR563543
3. F. Beukers, The zero-multiplicity of ternary recurrences. Compositio Math. 77 (1991), 165–177. Zbl0717.11012MR1091896
4. F. Beukers, R. Tijdeman, On the multiplicities of binary complex recurrences. Compositio Math. 51 (1984), 193–213. Zbl0538.10014MR739734
5. E. Bombieri, J. Müller, U. Zannier, Equations in one variable over function fields. Acta Arith. 99 (2001), 27–39. Zbl0973.11014MR1845361
6. Y. Bugeaud, K. Győry, Bounds for the solutions of unit equations. Acta Arith. 74 (1996), 67–80. Zbl0861.11023MR1367579
7. L. Cerlienco, M. Mignotte, F. Piras, Suites récurrentes linéaires: propriétés algébriques et arithmétiques. Enseign. Math. (2) 33 (1987), 67–108. Zbl0626.10008MR896384
8. W. D. Brownawell, D. Masser, Vanishing sums in function fields. Math. Proc. Cambridge Philos. Soc. 100 (1986), 427–434. Zbl0612.10010MR857720
9. J.-H. Evertse, On equations in two $S$-units over function fields of characteristic $0$. Acta Arith. 47 (1986), 233–253. Zbl0632.10015MR870667
10. J.-H. Evertse, K. Győry, On the number of solutions of weighted unit equations. Compositio Math. 66 (1988), 329–354. Zbl0644.10015MR948309
11. J.-H. Evertse, K. Győry, C. L. Stewart, R. Tijdeman, $S$-unit equations and their applications. In: New advances in transcendence theory (ed. by A. Baker), 110–174, Cambridge Univ. Press, Cambridge, 1988. Zbl0658.10023MR971998
12. J.-H. Evertse, H. P. Schlickewei, W. M. Schmidt, Linear equations in variables which lie in a multiplicative group. Ann. Math. 155 (2002), 1–30. Zbl1026.11038MR1923966
13. J.-H. Evertse, U. Zannier, Linear equations with unknowns from a multiplicative group in a function field. Preprint (http://www.math.leidenuniv.nl/~evertse/04-functionfields.ps). Zbl1185.11022
14. C. Fuchs, On the equation ${G}_{n}\left(x\right)={G}_{m}\left(P\left(x\right)\right)$ for third order linear recurring sequences. Port. Math. (N.S.) 61 (2004), 1–24. Zbl1114.11030MR2040240
15. C. Fuchs, A. Pethő, R. F. Tichy, On the Diophantine equation ${G}_{n}\left(x\right)={G}_{m}\left(P\left(x\right)\right)$. Monatsh. Math. 137 (2002), 173–196. Zbl1026.11039MR1942618
16. C. Fuchs, A. Pethő, R. F. Tichy, On the Diophantine equation ${G}_{n}\left(x\right)={G}_{m}\left(P\left(x\right)\right)$: Higher-order recurrences. Trans. Amer. Math. Soc. 355 (2003), 4657–4681. Zbl1026.11040MR1990766
17. C. Fuchs, A. Pethő, R. F. Tichy, On the Diophantine equation ${G}_{n}\left(x\right)={G}_{m}\left(y\right)$ with $Q\left(x,y\right)=0$. Preprint (http://finanz.math.tu-graz.ac.at/~fuchs/oegngmad4.ps). Zbl1215.11031
18. M. Laurent, Équations exponentielles polynômes et suites récurrentes linéares. Astérisque 147–148 (1987), 121–139. Zbl0621.10014MR891424
19. M. Laurent, Équations exponentielles-polynômes et suites récurrentes linéaires, II, J. Number Theory 31 (1989), 24-53. Zbl0661.10027MR978098
20. R. C. Mason, Equations over Function Fields. Lecture Notes in Math. 1068 (1984), Springer, Berlin, 149–157. Zbl0544.10015MR756091
21. R. C. Mason, Norm form equations I. J. Number Theory 22 (1986), 190–207. Zbl0578.10021MR826951
22. H. P. Schlickewei, Multiplicities of recurrence sequences. Acta Math. 176 (1996), 171–243 Zbl0880.11016MR1397562
23. H. P. Schlickewei, The multiplicity of binary recurrences. Invent. Math. 129 (1997), 11–36. Zbl0883.11008MR1464864
24. H. P. Schlickewei, W. M. Schmidt, The intersection of recurrence sequences. Acta Arith. 72 (1995), 1–44. Zbl0851.11007MR1346803
25. H. P. Schlickewei, W. M. Schmidt, The number of solutions of polynomial-exponential equations. Compositio Math. 120 (2000), 193–225. Zbl0949.11020MR1739179
26. W. M. Schmidt, The zero multiplicity of linear recurrence sequences. Acta Math. 182 (1999), 243–282. Zbl0974.11013MR1710183
27. W. M. Schmidt, Zeros of linear recurrence sequences. Publ. Math. Debrecen 56 (2000), 609–630. Zbl0963.11007MR1766002
28. W. M. Schmidt, Linear Recurrence Sequences and Polynomial-Exponential Equations. In: Diophantine Approximation (F. Amoroso, U. Zannier eds.), Proc. of the C.I.M.E. Conference, Cetraro (Italy) 2000, Springer-Verlag LNM 1819, 2003. Zbl1034.11011MR2009831
29. T. N. Shorey, R. Tijdeman, Exponential Diophantine Equations. Cambridge, Univ. Press, 1986. Zbl0606.10011MR891406
30. J. F. Voloch, Diagonal equations over function fields. Bol. Soc. Brasil. Mat. 16 (1985), 29–39. Zbl0612.10011MR847114
31. U. Zannier, Some remarks on the $S$-unit equation in function fields. Acta Arith. 64 (1993), 87–98. Zbl0786.11019MR1220487
32. U. Zannier, On the integer solutions of exponential equations in function fields. Ann. Inst. Fourier (Grenoble) 54 (2004), 849–874. Zbl1080.11028MR2111014

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