2,937 research outputs found
Algebraicity and transcendence of power series: combinatorial and computational aspects
DoctoralFrom ancient times, mathematicians are interested in the following question: is a given real number "algebraic" (that is, a root of a nonzero univariate polynomial with rational number coefficients), or is it "transcendental"? Although almost all real numbers are transcendental, it is notoriously difficult to actually prove, or disprove, the transcendence of a given constant. More recently, and especially with the advent of computers, different related questions arose: What is the "complexity" of a real number? How fast can one compute the first digits, or one single digit, of a (computable) real number? Can digits of algebraic numbers be computed faster than those of (computable) transcendental numbers? In this series of lectures, we will consider the (simpler) functional analogues of these questions: given a formal power series with rational number coefficients, decide whether it is algebraic (root of a nontrivial bivariate polynomial) or transcendental, and determine how fast can one compute its coefficients? We will first motivate these questions by presenting some examples of algebraic power series coming from combinatorics, with a focus on enumeration of lattice walks. Then we will discuss several methods that allow to discover and prove the nature (algebraic or transcendental) of a generating function, with an emphasis on an experimental mathematics approach combined with algorithmic methods such as Guess'n'Prove and Creative Telescoping. Finally, we will overview efficient algorithms for various operations on algebraic power series, including the computation of one or several selected terms
A probabilistic algorithm to test local algebraic observability in polynomial time
The following questions are often encountered in system and control theory.
Given an algebraic model of a physical process, which variables can be, in
theory, deduced from the input-output behavior of an experiment? How many of
the remaining variables should we assume to be known in order to determine all
the others? These questions are parts of the \emph{local algebraic
observability} problem which is concerned with the existence of a non trivial
Lie subalgebra of the symmetries of the model letting the inputs and the
outputs invariant. We present a \emph{probabilistic seminumerical} algorithm
that proposes a solution to this problem in \emph{polynomial time}. A bound for
the necessary number of arithmetic operations on the rational field is
presented. This bound is polynomial in the \emph{complexity of evaluation} of
the model and in the number of variables. Furthermore, we show that the
\emph{size} of the integers involved in the computations is polynomial in the
number of variables and in the degree of the differential system. Last, we
estimate the probability of success of our algorithm and we present some
benchmarks from our Maple implementation.Comment: 26 pages. A Maple implementation is availabl
On the non-holonomic character of logarithms, powers, and the n-th prime function
We establish that the sequences formed by logarithms and by "fractional"
powers of integers, as well as the sequence of prime numbers, are
non-holonomic, thereby answering three open problems of Gerhold [Electronic
Journal of Combinatorics 11 (2004), R87]. Our proofs depend on basic complex
analysis, namely a conjunction of the Structure Theorem for singularities of
solutions to linear differential equations and of an Abelian theorem. A brief
discussion is offered regarding the scope of singularity-based methods and
several naturally occurring sequences are proved to be non-holonomic.Comment: 13 page
On a conjecture by Pierre Cartier about a group of associators
In \cite{cartier2}, Pierre Cartier conjectured that for any non commutative
formal power series on with coefficients in a
\Q-extension, , subjected to some suitable conditions, there exists an
unique algebra homomorphism from the \Q-algebra generated by the
convergent polyz\^etas to such that is computed from
Drinfel'd associator by applying to each coefficient. We prove
exists and it is a free Lie exponential over . Moreover, we give a
complete description of the kernel of polyz\^eta and draw some consequences
about a structure of the algebra of convergent polyz\^etas and about the
arithmetical nature of the Euler constant
Subword complexity and Laurent series with coefficients in a finite field
Decimal expansions of classical constants such as , and
have long been a source of difficult questions. In the case of
Laurent series with coefficients in a finite field, where no carry-over
difficulties appear, the situation seems to be simplified and drastically
different. On the other hand, Carlitz introduced analogs of real numbers such
as , or . Hence, it became reasonable to enquire how
"complex" the Laurent representation of these "numbers" is. In this paper we
prove that the inverse of Carlitz's analog of , , has in general a
linear complexity, except in the case , when the complexity is quadratic.
In particular, this implies the transcendence of over \F_2(T). In the
second part, we consider the classes of Laurent series of at most polynomial
complexity and of zero entropy. We show that these satisfy some nice closure
properties
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