14,319 research outputs found
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
An Avoidance Principle with an Application to the Asymptotic Behaviour of Graded Local Cohomology
We present an Avoidance Principle for certain graded rings. As an application
we fill a gap in the proof of a result by Brodmann, Rohrer and Sazeedeh about
the antipolynomiality of the Hilbert-Samuel multiplicity of the graded
components of the local cohomology modules of a finitely generated module over
a Noetherian homogeneous ring with two-dimensional local base ring.Comment: 6 pages; to appear in Journal of Pure and Applied Algebra; corrected
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Stable domination and independence in algebraically closed valued fields
We seek to create tools for a model-theoretic analysis of types in
algebraically closed valued fields (ACVF). We give evidence to show that a
notion of 'domination by stable part' plays a key role. In Part A, we develop a
general theory of stably dominated types, showing they enjoy an excellent
independence theory, as well as a theory of definable types and germs of
definable functions. In Part B, we show that the general theory applies to
ACVF. Over a sufficiently rich base, we show that every type is stably
dominated over its image in the value group. For invariant types over any base,
stable domination coincides with a natural notion of `orthogonality to the
value group'. We also investigate other notions of independence, and show that
they all agree, and are well-behaved, for stably dominated types. One of these
is used to show that every type extends to an invariant type; definable types
are dense. Much of this work requires the use of imaginary elements. We also
show existence of prime models over reasonable bases, possibly including
imaginaries
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