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Displaying 61 –
80 of
859
We show how the size of the Galois groups of iterates of a quadratic polynomial f can be parametrized by certain rational points on the curves Cₙ: y² = fⁿ(x) and their quadratic twists (here fⁿ denotes the nth iterate of f). To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem...
We describe a sufficient condition for a finitely generated group to have infinite asymptotic dimension. As an application, we conclude that the first Grigorchuk group has infinite asymptotic dimension.
The theorem of Czerniakiewicz and Makar-Limanov, that all
the automorphisms of a free algebra of rank two are tame is proved here by
showing that the group of these automorphisms is the free product of two
groups (amalgamating their intersection), the group of all affine automorphisms
and the group of all triangular automorphisms. The method consists
in finding a bipolar structure. As a consequence every finite subgroup of
automorphisms (in characteristic zero) is shown to be conjugate to a group...
Currently displaying 61 –
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859