3,267 research outputs found
A dichotomy result for prime algebras of Gelfand-Kirillov dimension two
Let be an uncountable field. We show that a finitely generated prime
Goldie -algebra of quadratic growth is either primitive or satisfies a
polynomial identity, answering a question of Small in the affirmative.Comment: 10 page
Noetherian algebras over algebraically closed fields
Let be an uncountable algebraically closed field and let be a
countably generated left Noetherian -algebra. Then we show that is left Noetherian for any field extension of . We conclude that all
subfields of the quotient division algebra of a countably generated left
Noetherian domain over are finitely generated extensions of . We give
examples which show that need not remain left Noetherian if the
hypotheses are weakened.Comment: 10 page
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
A problem around Mahler functions
Let be a field of characteristic zero and and be two
multiplicatively independent positive integers. We prove the following result
that was conjectured by Loxton and van der Poorten during the Eighties: a power
series satisfies both a - and a -Mahler type functional
equation if and only if it is a rational function.Comment: 52 page
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