Eric Domenjoud

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We show in this note that the equation αx1 + #x22EF; +αxp≐ACβy1 + α +βyq where + is an AC operator and αx stands for x+...+x (α times), has exactly $$\left( { - 1} \right)^{p + q} \sum\limits_{i = 0}^p {\sum\limits_{j = 0}^q {\left( { - 1} \right)^{1 + 1} \left( {\begin{array}{*{20}c} p \\ i \\ \end{array} } \right)\left( {\begin{array}{*{20}c} q \\ j \\(More)
While connected rational arithmetical discrete lines and connected rational arithmetical discrete planes are entirely characterized, only partial results exist for the irrational arithmetical discrete planes. In the present paper, we focus on the connectedness of irrational arithmetical discrete planes, namely the arithmetical discrete planes with a normal(More)
We describe a new algorithm for solving a conjunction of linear diophantine equations, inequations and disequations in natural numbers. We derive our algorithm from one proposed by Elliott in 1903 for solving a single homogeneous equation. This algorithm was then extended to solve homogeneous systems of equations by MacMahon. We show how it further extends(More)
We investigate connections between a well known multidimensional continued fraction algorithm, the so-called fully subtractive algorithm, the finiteness property for β-numeration, and the connectedness of arithmetic discrete hyperplanes. A discrete hyperplane is said to be critical if its thickness is equal to the infimum of the set of thicknesses for which(More)