Resumen:
This thesis consists of two main theorems (Theorems 2 and 6) on the arithmetic
of polynomials and some consequences in Mathematical Logic.
A simple corollary of our first main theorem, inspired by a paper on powers
in arithmetic progressions by Hajdu [10], states in particular that given
any field F
the quantity a!+b, where a and b are coprime polynomials in F[t], cannot
be a power in F[t] for more than M = 4 distinct values of ! ! F, unless both
a and b have zero derivative.
Here by power, we mean a k-th power for some k " 2. This corollary was
actually our very first result and inspired the rest of the thesis. On the one
hand, one could naturally try to generalize it in the following ways:
1. Is the condition on coprimality really necessary? If not, how small
should the degree of the greatest common divisor of a and b be in order
to ensure the existence of an M (guessing that the more a and b have
factors in common, the bigger should be M);
2. What about other rings of functions? (such that subrings of function
fields, rings of analytic or meromorphic functions, . . . )
3. What about considering expressions of the form a!2 + b! + c instead
of a! + b? What about higher powers?
In the work presented here, we deal with generalizations of type 1 (Theorem
2) and of type 3 (Theorem 6).
On the other hand, the statement above and its possible generalizations
should say something about the first order language LP = {0, 1,+, P} (or
languages containing LP ), where P is a unary relation symbol and P(x) is
interpreted as “x is a power”.
Though to the best of our knowledge this language has not been previously
studied by logicians (though Macintyre’s language [12] is somewhat
related to it, as it contains a predicate Pk for each k " 2 interpreted as
“Pk(x) if and only if x is a k-th power”), the study of consecutive powers
and powers in arithmetic progression over the integers has a long history.
In 1640, Fermat conjectured that there does not exist four squares in arithmetic
progression, and this was proved later on by Euler.