20 research outputs found
The projective translation equation and rational plane flows. I
Let X=(x,y). A plane flow is a function F(X,t): R^2*R->R^2 such that
F(F(X,s),t)=F(X,s+t) for (almost) all real numbers x,y,s,t (the function F
might not be well-defined for certain x,y,t). In this paper we investigate
rational plane flows which are of the form F(X,t)=f(Xt)/t; here f is a pair of
rational functions in 2 real variables. These may be called projective flows,
and for a description of such flows only the knowledge of Cremona group in
dimension 1 is needed. Thus, the aim of this work is to completely describe
over R all rational solutions of the two dimensional translation equation
(1-z)f(X)=f(f(Xz)(1-z)/z). We show that, up to conjugation with a 1-homogenic
birational plane transformation (1-BIR), all solutions are as follows: a zero
flow, two singular flows, an identity flow, and one non-singular flow for each
non-negative integer N, called the level of the flow. The case N=0 stands
apart, while the case N=1 has special features as well. Conjugation of these
canonical solutions with 1-BIR produce a variety of flows with different
properties and invariants, depending on the level and on the conjugation
itself. We explore many more features of these flows; for example, there are 1,
4, and 2 essentially different symmetric flows in cases N=0, N=1, and N>=2,
respectively. Many more questions will be treated in the second part of this
work.Comment: 54 pages, 6 figures. Final version before proof
Multi-variable translation equation which arises from homothety
In many regular cases, there exists a (properly defined) limit of iterations
of a function in several real variables, and this limit satisfies the
functional equation (1-z)f(x)=f(f(xz)(1-z)/z); here z is a scalar and x is a
vector. This is a special case of a well-known translation equation. In this
paper we present a complete solution to this functional equation in case f is a
continuous function on a single point compactification of a 2-dimensional real
vector space. It appears that, up to conjugation by a homogeneous continuous
function, there are exactly four solutions. Further, in a 1-dimensional case we
present a solution with no regularity assumptions on f.Comment: 15 page
The Hardness of Code Equivalence over and its Application to Code-based Cryptography
International audienceThe code equivalence problem is to decide whether two linear codes over F_q are equivalent, that is identical up to a linear isometry of the Hamming space. In this paper, we review the hardness of code equivalence over F_q due to some recent negative results and argue on the possible implications in code-based cryptography. In particular, we present an improved version of the three-pass identification scheme of Girault and discuss on a connection between code equivalence and the hidden subgroup problem
On a Linear Functional Equation for Formal Power Series. Sitzungsberichte und Anzeiger der math.-nat. Klasse / Jahrgang 2001
Associative Formal Power Series in Two Indeterminates
Investigating the associativity equation for formal power series in two variables we show that the transcendental associative formal power series are of order one or two and that they can be represented by an invertible formal power series in one variable. We also discuss the convergence of associative formal power series