Riv.Mat.Univ.Parma (7) 3* (2004)
Exponential Diophantine equations and inequalities
Received: 29 February 2004
Mathematics Subject Classification (2000): 11D45 - 11D61
Let us consider the ring of power sums with algebraic
and positive integral roots, i.e. of functions of N of
a(n) = b1cn
1 + b2cn
2 + ... + bhcn
with biÎ Q0>
and c1 > c2 >... > ch Î Z+.
Since long ago, Diophantine equations and inequalities involving
power sums have been studied using the estimates for linear forms
logarithms due to A. Baker, but many problems remained unsolved.
Recently P. Corvaja and U. Zannier have found new results on
problems by a different method, applying in this context the
Theorem by W. Schmidt.
Here we will first have a review on some of such results,
will show some recent results obtained by the author, partially
Fuchs. We will first deal with the finiteness of the solutions
(n,y) Î NxZ
of the equation
F(a(n),y) = f(n),
where FÎQ[X,Y] is monic,
absolutely irreducible of degree at least 2,
and f ÎZ[X] and the power
sum a are not constant.
Then we will consider equations and inequalities where
sums are involved as, for example, the equation
a0(n)yd + a1(n)yd-1 + ...
+ a d-1(n)y + ad(n) = 0
and the inequality
where a0, ..., ad are power sums,
e > 0, F is monic in y
and a is a
quantity depending on the dominant roots of the power sums
as coefficients in F. We will show that, under suitable
all the solutions of (2), y can be parametrized by some power sum
finite set. We will reach a similar conclusion also for (3).
All these results generalize some results by P. Corvaja and
on such problems.