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On The p‐Adic Integral of An Exponential Polynomial

On The p‐Adic Integral of An Exponential Polynomial ON THE /7-ADIC INTEGRAL OF AN EXPONENTIAL POLYNOMIAL G. R. EVEREST In their recent paper [8], Shparlinski and van der Poorten prove that an algebraic linear recurrence sequence takes a mean /?-adic value as the argument runs over integer values. As we shall indicate below, this value is natural, in the sense that it admits a /j-adic integral representation. Also, this value is of arithmetic interest. The results in [8] have spawned the pursuit of mean value theorems in other contexts. For example, the Norm-Form Equation [4], and the perennial occupation with the values taken by sums of S-units in algebraic number fields [3]. In this paper we aim to put the whole question of the existence of mean values on a properly general footing. We shall attempt to highlight the arithmetic nature of the mean values, where they are known to exist. Also, we shall consider the effective nature of the results. To this end, let E(x) denote a generalized exponential polynomial in the variable x = (x ,...,x ). This is an expression of the form 1 n EQO - E*=i A (x)<x*t •••«£?• Here, the cc are algebraic numbers and yi (x)6Q[x] = t tj ( http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

On The p‐Adic Integral of An Exponential Polynomial

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/27.4.334
Publisher site
See Article on Publisher Site

Abstract

ON THE /7-ADIC INTEGRAL OF AN EXPONENTIAL POLYNOMIAL G. R. EVEREST In their recent paper [8], Shparlinski and van der Poorten prove that an algebraic linear recurrence sequence takes a mean /?-adic value as the argument runs over integer values. As we shall indicate below, this value is natural, in the sense that it admits a /j-adic integral representation. Also, this value is of arithmetic interest. The results in [8] have spawned the pursuit of mean value theorems in other contexts. For example, the Norm-Form Equation [4], and the perennial occupation with the values taken by sums of S-units in algebraic number fields [3]. In this paper we aim to put the whole question of the existence of mean values on a properly general footing. We shall attempt to highlight the arithmetic nature of the mean values, where they are known to exist. Also, we shall consider the effective nature of the results. To this end, let E(x) denote a generalized exponential polynomial in the variable x = (x ,...,x ). This is an expression of the form 1 n EQO - E*=i A (x)<x*t •••«£?• Here, the cc are algebraic numbers and yi (x)6Q[x] = t tj (

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Jul 1, 1995

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