**AMEDEO SCREMIN**
* Exponential Diophantine equations and inequalities
*

Mathematics Subject Classification (2000):

**Abstract**:
Let us consider the ring of power sums with algebraic
coefficients
and positive integral roots, i.e. of functions of** N** of
the form

(1)
a(n) = b_{1}c^{n}_{
1} + b_{2}c^{n}_{
2} + ... + b_{h}c^{n}
_{h},

with b_{i}Î ** Q
**
and c

(2)
a_{0}(n)y^{d} + a_{1}(n)y^{d-1} + ...
+ a _{d-1}(n)y + a_{d}(n) = 0

and the inequality

(3)
|F(n,y)|< a^{(n-d-1-e)},

where a_{0}, ..., a_{d} are power sums,
e > 0, F is monic in y
and a is a
quantity depending on the dominant roots of the power sums
appearing
as coefficients in F. We will show that, under suitable
assumptions, for
all the solutions of (2), y can be parametrized by some power sum
in a
finite set. We will reach a similar conclusion also for (3).

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