Integral Specialization of families of
rational functions
The Hilbert-Siegel problem:
This to determine the indecomposable
polynomials h(x) \in Q[x] for which h(x)-z is reducible (in Q[x])
for an infinite number of z
in Z \ {h(Q)
∩ Z}. Call this having o-value reducibility (o standing
for "outside"). This paper shows there exists a family of
polynomials of degree 5, having o-value reducibility. This completes
the results in Thm. 2.6 of Variables separated polynomials and Moduli Spaces which showed the only possible example would be where the degree is 5 . The last paper also contains definitive results for other number fields, variation of other coefficients, and comments on P. Mueller's generalization replacing h(x)-z=0
by any genus 0 equation.
Ingredients of the Proof:
The complete reduction of the problem has many steps. One essential is
results about the possible monodromy groups of such polynomials h that used the classification of
finite simple groups. We use Hurwitz spaces to study the families of
polynomials that could be exceptions. The first business is to
show explicitly that the corresponding Hurwitz spaces are uni-rational
varieties. Then, the paper locates rational points on these spaces that
would produce the exceptional degree 5 polynomials. It is a piece of
lucky arithmetic, requiring a delicate calculation to find the
required points.
General Context and Other Results:
We also gives the problem a general context by considering rational
functions (Siegel version) that could possibly have this property and
by dropping the limitation that z
is in a fractional ideal
(Néron version). Certain geometric conditions must be satisfied
for either. We give a large set of Hurwitz families that could contain
members that satisfy the Siegel or Néron versions of the
problem. These examples challenge how to draw arithmetic conclusions
akin to this paper's about curves in a Hurwitz family without depending
on effectively parametrizing the families.