202 research outputs found
Finite automata and algebraic extensions of function fields
We give an automata-theoretic description of the algebraic closure of the
rational function field F_q(t) over a finite field, generalizing a result of
Christol. The description takes place within the Hahn-Mal'cev-Neumann field of
"generalized power series" over F_q. Our approach includes a characterization
of well-ordered sets of rational numbers whose base p expansions are generated
by a finite automaton, as well as some techniques for computing in the
algebraic closure; these include an adaptation to positive characteristic of
Newton's algorithm for finding local expansions of plane curves. We also
conjecture a generalization of our results to several variables.Comment: 40 pages; expanded version of math.AC/0110089; v2: refereed version,
includes minor edit
Shift Radix Systems - A Survey
Let be an integer and . The {\em shift radix system} is defined by has the {\em finiteness
property} if each is eventually mapped to
under iterations of . In the present survey we summarize
results on these nearly linear mappings. We discuss how these mappings are
related to well-known numeration systems, to rotations with round-offs, and to
a conjecture on periodic expansions w.r.t.\ Salem numbers. Moreover, we review
the behavior of the orbits of points under iterations of with
special emphasis on ultimately periodic orbits and on the finiteness property.
We also describe a geometric theory related to shift radix systems.Comment: 45 pages, 16 figure
Comments on the height reducing property II
A complex number αα is said to satisfy the height reducing property if there is a finite set F⊂ZF⊂Z such that Z[α]=F[α]Z[α]=F[α], where ZZ is the ring of the rational integers. It is easy to see that αα is an algebraic number when it satisfies the height reducing property. We prove the relation Card(F)≥max{2,|Mα(0)|}Card(F)≥max{2,|Mα(0)|}, where MαMα is the minimal polynomial of αα over the field of the rational numbers, and discuss the related optimal cases, for some classes of algebraic numbers αα. In addition, we show that there is an algorithm to determine the minimal height polynomial of a given algebraic number, provided it has no conjugate of modulus one
Fractal tiles associated with shift radix systems
Shift radix systems form a collection of dynamical systems depending on a
parameter which varies in the -dimensional real vector space.
They generalize well-known numeration systems such as beta-expansions,
expansions with respect to rational bases, and canonical number systems.
Beta-numeration and canonical number systems are known to be intimately related
to fractal shapes, such as the classical Rauzy fractal and the twin dragon.
These fractals turned out to be important for studying properties of expansions
in several settings. In the present paper we associate a collection of fractal
tiles with shift radix systems. We show that for certain classes of parameters
these tiles coincide with affine copies of the well-known tiles
associated with beta-expansions and canonical number systems. On the other
hand, these tiles provide natural families of tiles for beta-expansions with
(non-unit) Pisot numbers as well as canonical number systems with (non-monic)
expanding polynomials. We also prove basic properties for tiles associated with
shift radix systems. Indeed, we prove that under some algebraic conditions on
the parameter of the shift radix system, these tiles provide
multiple tilings and even tilings of the -dimensional real vector space.
These tilings turn out to have a more complicated structure than the tilings
arising from the known number systems mentioned above. Such a tiling may
consist of tiles having infinitely many different shapes. Moreover, the tiles
need not be self-affine (or graph directed self-affine)
Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)
Oxford, UK, 26 August 200
Decidability Problems for Self-induced Systems Generated by a Substitution
International audienceIn this talk we will survey several decidability and undecidability results on topological properties of self-affine or self-similar fractal tiles. Such tiles are obtained as fixed point of set equations governed by a graph. The study of their topological properties is known to be complex in general: we will illustrate this by undecidability results on tiles generated by multitape automata. In contrast, the class of self affine tiles called Rauzy fractals is particularly interesting. Such fractals provide geometrical representations of self-induced mathematical processes. They are associated to one-dimensional combinatorial substitutions (or iterated morphisms). They are somehow ubiquitous as self-replication processes appear naturally in several fields of mathematics. We will survey the main decidable topological properties of these specific Rauzy fractals and detail how the arithmetic properties yields by the combinatorial substitution underlying the fractal construction make these properties decidable. We will end up this talk by discussing new questions arising in relation with continued fraction algorithm and fractal tiles generated by S-adic expansion systems
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