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DIOPHANTINE EQUATIONS FOR POLYNOMIALS WITH RESTRICTED COEFFICIENTS, I (POWER VALUES)

Part of: Diophantine equations Algebraic number theory: global fields

Published online by Cambridge University Press:  22 February 2022

LAJOS HAJDU Open the ORCID record for LAJOS HAJDU [Opens in a new window]  and
NÓRA VARGA Open the ORCID record for NÓRA VARGA [Opens in a new window]
LAJOS HAJDU
Affiliation:
Institute of Mathematics, University of Debrecen, P.O. Box 400, H-4002 Debrecen, Hungary and Alfréd Rényi Institute of Mathematics, Budapest P.O. Box 127, H-1364, Hungary e-mail: hajdul@science.unideb.hu, hajdu@renyi.hu
NÓRA VARGA*
Affiliation:
Institute of Mathematics, University of Debrecen, P.O. Box 400, H-4002 Debrecen, Hungary and MTA-DE Research Group ‘Equations, Functions and Curves’, Eötvös Loránd Research Network (ELKH), Hungary
*
e-mail: nvarga@science.unideb.hu
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Abstract

We give effective finiteness results for the power values of polynomials with coefficients composed of a fixed finite set of primes; in particular, of Littlewood polynomials.

Keywords

integers composed of fixed primespolynomials with restricted coefficientshyperelliptic and superelliptic Diophantine equationsthe Schinzel–Tijdeman equation

MSC classification

Primary: 11D41: Higher degree equations; Fermat's equation
Secondary: 11R09: Polynomials (irreducibility, etc.)

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 2 , October 2022 , pp. 254 - 263
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Dedicated to the memory of Andrzej Schinzel

Research supported in part by the Eötvös Loránd Research Network (ELKH), by the NKFIH grants 115479, 128088 and 130909, and the project EFOP-3.6.1-16-2016-00022 co-financed by the European Union and the European Social Fund.

References

Bloch, A. and Pólya, G., ‘On the roots of certain algebraic equations’, Proc. Lond. Math. Soc. (3) 33 (1932), 102–114.CrossRefGoogle Scholar
Borwein, P. and Erdélyi, T., Polynomials and Polynomial Inequalities (Springer-Verlag, New York, 1995).CrossRefGoogle Scholar
Borwein, P. and Erdélyi, T., ‘On the zeros of polynomials with restricted coefficients’, Illinois J. Math. 41 (1997), 667–675.CrossRefGoogle Scholar
Borwein, P. and Mossinghoff, M. J., ‘Polynomials with height 1 and prescribed vanishing at 1’, Exp. Math. 9 (2000), 425–433.CrossRefGoogle Scholar
Brindza, B., ‘On $S$ -integral solutions of the equation ${y}^m=f(x)$ ’, Acta Math. Hungar. 44 (1984), 133–139.CrossRefGoogle Scholar
Drungilas, P. and Dubickas, A., ‘Roots of polynomials of bounded height’, Rocky Mountain J. Math. 39 (2009), 527–543.CrossRefGoogle Scholar
Dubickas, A. and Jankauskas, J., ‘On Newman polynomials which divide no Littlewood polynomial’, Math. Comp. 78 (2009), 327–344.CrossRefGoogle Scholar
Erdős, P. and Turán, P., ‘On the distribution of roots of polynomials’, Ann. of Math. (2) 57 (1950), 105–119.CrossRefGoogle Scholar
Evertse, J.-H. and Győry, K., Unit Equations in Diophantine Number Theory (Cambridge University Press, Cambridge, 2015).CrossRefGoogle Scholar
Győry, K., ‘Sur l’irréducibilité d’une classe des polynômes, II’, Publ. Math. Debrecen 19 (1972), 293–326.CrossRefGoogle Scholar
Mossinghoff, M. J., ‘Polynomials with restricted coefficients and prescribed noncyclotomic factors’, LMS J. Comput. Math. 6 (2003), 314–325.CrossRefGoogle Scholar
Peled, R., Sen, A. and Zeitouni, O., ‘Double roots of random Littlewood polynomials’, Israel J. Math. 213 (2016), 55–77.CrossRefGoogle Scholar
Schinzel, A. and Tijdeman, R., ‘On the equation ${y}^m=P(x)$ ’, Acta Arith. 31 (1976), 199–204.CrossRefGoogle Scholar
Schur, I., ‘Untersuchungen über algebraische Gleichungen’, Sitz. Preuss. Akad. Wiss., Phys.-Math. Kl. 7–10(1933), 403–428.Google Scholar
Shorey, T. N. and Tijdeman, R., Exponential Diophantine Equations (Cambridge University Press, Cambridge, 1986).CrossRefGoogle Scholar
Szegő, G., ‘Bemerkungen zu einem Satz von E. Schmidt über algebraische Gleichungen’, Sitz. Preuss. Akad. Wiss., Phys.-Math. K1. 8(1934), 86–98.Google Scholar
Tijdeman, R., ‘Applications of the Gel’fond–Baker method to rational number theory’, in: Topics in Number Theory (Proceedings of the Conference at Debrecen 1974), Colloquia Mathematica Societatis János Bolyai, 13 (ed P. Turán) (North-Holland, Amsterdam, 1976), 399–416.Google Scholar