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 Fq\mathbf{F}_q 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

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