39 research outputs found
Rado's theorem for rings and modules
We extend classical results of Rado on partition regularity of systems of
linear equations with integer coefficients to the case when the coefficient
ring is either an arbitrary domain or a noetherian ring. The crucial idea is to
study partition regularity for general modules rather than only for rings.
Contrary to previous techniques, our approach is independent of the
characteristic of the coefficient ring.Comment: 19 page
Which weakly ramified group actions admit a universal formal deformation?
Consider a formal (mixed-characteristic) deformation functor D of a
representation of a finite group G as automorphisms of a power series ring
k[[t]] over a perfect field k of positive characteristic. Assume that the
action of G is weakly ramified, i.e., the second ramification group is trivial.
Examples of such representations are provided by a group action on an ordinary
curve: the action of a ramification group on the completed local ring of any
point on such a curve is weakly ramified.
We prove that the only such D that are not pro-representable occur if k has
characteristic two and G is of order two or isomorphic to a Klein group.
Furthermore, we show that only the first of those has a non-pro-representable
equicharacteristic deformation functor.Comment: 16 pages; further minor correction
Simultaneous -orderings and minimising volumes in number fields
In the paper "On the interpolation of integer-valued polynomials" (Journal of
Number Theory 133 (2013), pp. 4224--4232.) V. Volkov and F. Petrov consider the
problem of existence of the so-called -universal sets (related to
simultaneous -orderings of Bhargava) in the ring of Gaussian integers. We
extend their results to arbitrary imaginary quadratic number fields and prove
an existence theorem that provides a strong counterexample to a conjecture of
Volkov-Petrov on minimal cardinality of -universal sets. Along the way, we
discover a link with Euler-Kronecker constants and prove a lower bound on
Euler-Kronecker constants which is of the same order of magnitude as the one
obtained by Ihara.Comment: new version, substantial corrections in section 6, will appear in
Journal of Number Theor
Substitutive systems and a finitary version of Cobham's theorem
We study substitutive systems generated by nonprimitive substitutions and
show that transitive subsystems of substitutive systems are substitutive. As an
application we obtain a complete characterisation of the sets of words that can
appear as common factors of two automatic sequences defined over
multiplicatively independent bases. This generalises the famous theorem of
Cobham.Comment: 23 pages. v2: incorporates referee's comments, updated references, to
appear in Combinatoric
Multiband linear cellular automata and endomorphisms of algebraic vector groups
We propose a correspondence between certain multiband linear cellular
automata - models of computation widely used in the description of physical
phenomena - and endomorphisms of certain algebraic unipotent groups over finite
fields. The correspondence is based on the construction of a universal element
specialising to a normal generator for any finite field. We use this
correspondence to deduce new results concerning the temporal dynamics of such
automata, using our prior, purely algebraic, study of the endomorphism ring of
vector groups. These produce 'for free' a formula for the number of fixed
points of the -iterate in terms of the -adic valuation of , a
dichotomy for the Artin-Mazur dynamical zeta function, and an asymptotic
formula for the number of periodic orbits. Since multiband linear cellular
automata simulate higher order linear automata (in which states depend on
finitely many prior temporal states, not just the direct predecessor), the
results apply equally well to that class.Comment: 11 page