163 research outputs found
F-sets and finite automata
The classical notion of a k-automatic subset of the natural numbers is here
extended to that of an F-automatic subset of an arbitrary finitely generated
abelian group equipped with an arbitrary endomorphism F. This is
applied to the isotrivial positive characteristic Mordell-Lang context where F
is the Frobenius action on a commutative algebraic group G over a finite field,
and is a finitely generated F-invariant subgroup of G. It is shown
that the F-subsets of introduced by the second author and Scanlon are
F-automatic. It follows that when G is semiabelian and X is a closed subvariety
then X intersect is F-automatic. Derksen's notion of a k-normal subset
of the natural numbers is also here extended to the above abstract setting, and
it is shown that F-subsets are F-normal. In particular, the X intersect
appearing in the Mordell-Lang problem are F-normal. This generalises
Derksen's Skolem-Mahler-Lech theorem to the Mordell-Lang context.Comment: The final section is revised following an error discovered by
Christopher Hawthorne; it is no longer claimed that an F-normal subset has a
finite symmetric difference with an F-subset. The main theorems of the paper
remain unchange
inverse orbit generating functions almost always have natural boundaries
The function sends to resp. according
as is odd, resp. even, where . The map
sends integers to integers, and for let mean that is in the forward orbit of under iteration of
We consider the generating functions which are holomorphic in the unit disk. We give
sufficient conditions on for the functions have the unit
circle as a natural boundary to analytic continuation. For the
function these conditions hold for all to show that
has the unit circle as a natural boundary except possibly for and . The Conjecture is equivalent to the assertion that
is a rational function of for the remaining values .Comment: 15 page
On the set of zero coefficients of a function satisfying a linear differential equation
Let be a field of characteristic zero and suppose that satisfies a recurrence of the form
for sufficiently large, where are polynomials in
. Given that is a nonzero constant polynomial, we show that the
set of for which is a union of finitely many
arithmetic progressions and a finite set. This generalizes the
Skolem-Mahler-Lech theorem, which assumes that satisfies a linear
recurrence. We discuss examples and connections to the set of zero coefficients
of a power series satisfying a homogeneous linear differential equation with
rational function coefficients.Comment: 11 page
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