Polynomials with roots modulo every integer

Berend, Daniel; Bilu, Yuri

doi:10.1090/S0002-9939-96-03210-8

Polynomials with roots modulo every integer
HTML articles powered by AMS MathViewer

by Daniel Berend and Yuri Bilu PDF

Proc. Amer. Math. Soc. 124 (1996), 1663-1671 Request permission

Abstract:

Given a polynomial with integer coefficients, we calculate the density of the set of primes modulo which the polynomial has a root. We also give a simple criterion to decide whether or not the polynomial has a root modulo every non-zero integer.

References

James Ax, Solving diophantine problems modulo every prime, Ann. of Math. (2) 85 (1967), 161–183. MR 209224, DOI 10.2307/1970438
Eric Bach, Explicit bounds for primality testing and related problems, Math. Comp. 55 (1990), no. 191, 355–380. MR 1023756, DOI 10.1090/S0025-5718-1990-1023756-8
M. Boshernitzan, On recurrence and spread in dynamical systems, preprint.
D. Berend and J. E. Harmse, On polynomial-factorial and similar diophantine equations, preprint.
D. Berend and C. F. Osgood, On the equation $P(x)=n!$ and a question of Erdős, J. Number Theory 42 (1992), 189–193.
Z. I. Borevich and I. R. Shafarevich, J. Number Theory, Academic Press, New York, 1966.
J. W. S. Cassels and A. Fröhlich, Algebraic Number Theory (Proceedings of an Instructional Conference, University of Sussex, 1965),, Academic Press and St Edmundsbury Press, Suffolk, 1990.
P. Erdős and R. Obláth, Über diophantische Gleichungen der Form $n!=x^{p}\pm y^{p}$ und $n!\pm m!=x^{p}$, Acta Szeged 8 (1937), 241–255.
M. Fried and G. Sacerdote, Solving Diophantine problems over all residue class fields of a number field and all finite fields, Ann. of Math. (2) 104 (1976), no. 2, 203–233. MR 491477, DOI 10.2307/1971045
H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625
Richard K. Guy, Unsolved problems in number theory, Problem Books in Mathematics, Springer-Verlag, New York-Berlin, 1981. MR 656313
T. Kamae and M. Mendès France, van der Corput’s difference theorem, Israel J. Math. 31 (1978), no. 3-4, 335–342. MR 516154, DOI 10.1007/BF02761498
L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0419394
J. C. Lagarias, Sets of primes determined by systems of polynomial congruences, Illinois J. Math. 27 (1983), no. 2, 224–239. MR 694641
J. C. Lagarias, H. L. Montgomery, and A. M. Odlyzko, A bound for the least prime ideal in the Chebotarev density theorem, Invent. Math. 54 (1979), no. 3, 271–296. MR 553223, DOI 10.1007/BF01390234
J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 409–464. MR 0447191
Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers, 2nd ed., Springer-Verlag, Berlin; PWN—Polish Scientific Publishers, Warsaw, 1990. MR 1055830
J. Osterlé, Versions effectives du théorème de Chebotarev sous l’hypothése de Riemann généralisé,, Soc. Math. France Astérisque 61 (1979), 165–167.
A. Sárkőzy, On difference sets of sequences of integers. I, Acta Math. Acad. Sci. Hungar. 31 (1978), no. 1-2, 125–149. MR 466059, DOI 10.1007/BF01896079
A. Sárközy, On difference sets of sequences of integers. II, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 21 (1978), 45–53 (1979). MR 536201
A. Sárközy, On difference sets of sequences of integers. II, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 21 (1978), 45–53 (1979). MR 536201
Wolfgang M. Schmidt, Construction and estimation of bases in function fields, J. Number Theory 39 (1991), no. 2, 181–224. MR 1129568, DOI 10.1016/0022-314X(91)90044-C
Volker Schulze, Die Primteilerdichte von ganzzahligen Polynomen. I, J. Reine Angew. Math. 253 (1972), 175–185 (German). MR 299592, DOI 10.1515/crll.1972.253.175
Volker Schulze, Die Primteilerdichte von ganzzahligen Polynomen. II, J. Reine Angew. Math. 256 (1972), 153–162 (German). MR 311616, DOI 10.1515/crll.1972.256.153
Volker Schulze, Die Primteilerdichte von ganzzahligen Polynomen. III, J. Reine Angew. Math. 273 (1975), 144–145 (German). MR 369324, DOI 10.1515/crll.1975.273.144
B.L. van der Waerden, Algebra I, Springer, 1971.