Counting the values taken by algebraic exponential polynomials

Everest, G.; Shparlinski, I.

doi:10.1090/S0002-9939-99-04728-0

Counting the values taken by algebraic exponential polynomials
HTML articles powered by AMS MathViewer

by G. R. Everest and I. E. Shparlinski PDF

Proc. Amer. Math. Soc. 127 (1999), 665-675 Request permission

Abstract:

We prove an effective mean-value theorem for the values of a non-degenerate, algebraic exponential polynomial in several variables. These objects generalise simultaneously the fundamental examples of linear recurrence sequences and sums of $S$-units. The proof is based on an effective, uniform estimate for the deviation of the exponential polynomial from its expected value. This estimate is also used to obtain a non-effective asymptotic formula counting the norms of these values below a fixed bound.

References

G. R. Everest, Counting the values taken by sums of $S$-units, J. Number Theory 35 (1990), no. 3, 269–286. MR 1062335, DOI 10.1016/0022-314X(90)90118-B
G. R. Everest, On the $p$-adic integral of an exponential polynomial, Bull. London Math. Soc. 27 (1995), no. 4, 334–340. MR 1335283, DOI 10.1112/blms/27.4.334
G. R. Everest and I. E. Shparlinski, Divisor sums of generalised exponential polynomials, Canad. Math. Bull. 39 (1996), no. 1, 35–46. MR 1382488, DOI 10.4153/CMB-1996-005-5
Jan-Hendrik Evertse, On sums of $S$-units and linear recurrences, Compositio Math. 53 (1984), no. 2, 225–244. MR 766298
Serge Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0282947
A. J. van der Poorten and H.-P. Schlickewei, The growth conditions for recurrence sequences, Macquarie Math. Reports 82-0041 (1982).
A. J. van der Poorten and I. Shparlinski, On the number of zeros of exponential polynomials and related questions, Bull. Austral. Math. Soc. 46 (1992), 401-412.
H. P. Schlickewei and Wolfgang M. Schmidt, On polynomial-exponential equations, Math. Ann. 296 (1993), no. 2, 339–361. MR 1219906, DOI 10.1007/BF01445109
—, On polynomial-exponential equations, II (to appear).
Klaus Schmidt and Tom Ward, Mixing automorphisms of compact groups and a theorem of Schlickewei, Invent. Math. 111 (1993), no. 1, 69–76. MR 1193598, DOI 10.1007/BF01231280
Wolfgang M. Schmidt, Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, vol. 1467, Springer-Verlag, Berlin, 1991. MR 1176315, DOI 10.1007/BFb0098246
T. N. Shorey and R. Tijdeman, Exponential Diophantine equations, Cambridge Tracts in Mathematics, vol. 87, Cambridge University Press, Cambridge, 1986. MR 891406, DOI 10.1017/CBO9780511566042
I. Shparlinski, On the number of distinct prime divisors of recurrence sequences, Matem. Zametki 42 (1987), 494–507 (Russian).