A second tool is p-adic analysis [29], [131], [620]; in particular Strassmann's
Theorem [1261], sometimes called the p-adic Weierstrass Preparation Theorem.
Section 1.2 provides a basic introduction to this beautiful theory. At several points
in the text, results about recurrence sequences will be given where the most natural
proofs seem to come from p-adic analysis. We can offer no explanation for this
phenomenon. For example, in Section 1.2, we give a simple proof of a special
case of the Hadamard quotient problem using p-adic analysis. The general case
has now been resolved and the methods are still basically p-adic. Similarly, when
it is applicable, p-adic analysis produces very good estimates for the number of
solutions of equations; compare the estimate of [1123] based on new results on
5-unit equations with that of [1038] obtained by the p-adie method. On the other
hand, a disadvantage of this approach is its apparent non-effectiveness in estimating
the size of solutions.
The simple observation that any field of zero characteristic over which a linear
recurrence sequence is defined may be assumed to be finitely generated over Q will
be used repeatedly. Indeed, it is enough to consider the field obtained from Q by
adjoining the initial values and the coefficients of the characteristic polynomial.
Then, using specialization arguments [1026] and [1037], we may restrict ourselves
to studying sequences over an algebraic extension of Qp or even just over Qp,
using a nice idea of Cassels [213]. Cassels shows that given any field F, finitely
generated over Q, and any finite subset M G F, there exist infinitely many rational
primes p such that there is an embedding (p : F Qp with ordp (/?(/i) = 0 for all
fi G M. A critical feature is that the embedding is into Qp, rather than a 'brute
force' embedding into an algebraic extension of Qp. The upshot is that for many
natural problems over general fields of zero characteristic, one can expect to get
results that are not worse than the corresponding one in the algebraic number field
case, or even for the case of rational numbers. Moreover, there are a number of
examples in the case of function fields where even stronger results can be obtained,
see [128], [160], [167], [171], [548], [781], [871] [920], [1002], [1041], [1162],
[1308], [1309], [1324], [1373].
Thirdly, many results depend on bounds for linear forms in the logarithms of
algebraic numbers. Section 1.3 gives an indication of the connection between the
theory of linear recurrence sequences and linear forms in logarithms by consider-
ing the apparently simple question: How quickly does a linear recurrence sequence
grow? After the first results of Baker [50], [51], [52], [53], [54], [55], and their p-
adic generalizations, for example those of van der Poorten [1017], a vast number of
further results, generalizations and improvements have been obtained; appropriate
references can be found in [1324]. For our purposes, the modern sharper bounds
do not imply any essentially stronger results than those relying on [55] and [1017].
In certain cases more recent results do allow the removal of some logarithmic terms;
[1369] is an example. We mostly content ourselves with consequences of the rela-
tively old results.
Fourthly and finally, several results on growth rate estimates or zero multi-
plicity are based upon properties of sums of 5-units. Specifically, linear recurrence
sequences provide a special case of 5-unit sums. Section 1.5 gives a basic account of
the way results about sums of 5-units can be applied to linear recurrence sequences.
This does not do justice to the full range of applicability of results about sums of
5-units applications will reverberate throughout the text.
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