Searcharxiv⌕ Search

arXiv subjects

Roland Bacher

Publications and source records attributed to Roland Bacher.

54 records · Page 3Linked to original sources
arXiv2006

Sur le groupe d'interpolation

We study the interpolation group whose elements are suitable pairs of formal power series. This group has a faithful representation into infinite lower triangular matrices and carries thus a natural structure as a Lie group. The matrix exponential between its Lie algebra and its matrix representation gives rise to a function with interesting properties extending the usual exponential function to two variables (which are formal power series) We finish with an application to enumerative combinatorics and the description of an algebra which generalizes the interpolation group.

math.CO↗
arXiv2006

La factorisation de $x^{p^l}-x\in{\mathbb F}\_p[x]$ selon la trace

We present a few factorizations of polynomials over finite fields. These factorizations are related to traces, compositions of polynomials and binomial coefficients. As a corollary we obtain a description of all irreducible polynomials $Q\in {\mathbb F}\_2[x]$ such that $Q(1+x+x^2)$ (or $Q(x+x^2)$) remain irreducible.

math.NT↗
arXiv2005

Spindle configurations of skew lines

We prove a conjecture of Crapo and Penne which characterizes isotopy classes of skew configurations with spindle-structure. We use this result in order to define an invariant, spindle-genus, for spindle-configurations. We also slightly simplify the exposition of some known invariants for configurations of skew lines and use them to define a natural partition of the lines in a skew configuration. Finally, we describe an algorithm which constructs a spindle in a given switching class, or proves non-existence of such a spindle.

math.GT↗
arXiv2004

On generating series of complementary planar trees

We generalize and reprove an identity of Parker and Loday. It states that certain pairs of generating series associated to pairs of labelled rooted planar trees are mutually inverse under composition.

math.CO↗
arXiv2004

On Minkowksi's bound for lattice-packings

We give a new proof of the Minkowski-Hlawka bound on the existence of dense lattices. The proof is based on an elementary method for constructing dense lattices which is almost effective.

math.NT↗
arXiv2003

Counting Triangulations of Configurations

After defining convex near-polygons, a formula enumerating the number of triangulations of such configurations is derived in terms of edge-polynomials. The paper describes also a transfer-matrix approach for computing quantities related to triangulations.

math.CO↗
arXiv2003

Le cocycle du verger

We define and prove uniqueness of a natural homomorphism (called the Orchard morphism) from some groups associated naturally to a finite set $E$ to the group ${\mathcal E}(E)$ of two-partitions of $E$ representing equivalence relations having at most two classes on $E$. As an application, given a finite generic configuration ${\mathcal C}\subset {\mathbf R}^d$, we exhibit a natural partition of ${\mathcal C}$ in two sets.

math.CO↗
arXiv2003

The Orchard Morphism

We describe and prove uniqueness of a natural homomorphism between some groups associated to finite sets.

math.CO↗
arXiv2003

Symmetric Pascal matrices modulo p

We study characteristic polynomials of symmetric matrices with entries ${i+j\choose i}$ the binomial coefficients, over finite fields.

math.NT↗
arXiv2002

Chromatic properties of generic planar configurations of points

We study the Orchard relation for generic configurations of points in the plane (also called order types). We introduce infinitesimally-close points and analyse the relation of this notion with the Orchard relation. The second part of the paper deals with monochromatic configurations (for the Orchard relation). We give the complete list of all monochromatic configurations up to 7 points and present some constructions and families of monochromatic configurations.

math.GT↗
arXiv2002

An Orchard Theorem

We describe a natural way to plant cherry- and plumtrees at prescribed generic locations in an orchard.

math.CO↗
arXiv2002

Matrices autosimilaires

We describe a class of matrices whose determinants are trivial to compute. A nice example of such a matrix is given by considering the symmetric matrix with entries {i+j choose i} (mod 2) in {0,1}, 0 <= i,j < n the binomial coefficients modulo 2 represented by {0,1}. The determinant of the above matrix is then closely related to the Thue-Morse sequence counting digits modulo 2 of binary integers.

math.RA↗
arXiv2001

Counting 1-vertex Triangulations Of Oriented Surfaces

A {\em $1-$vertex triangulation} of an oriented compact surface $S$ of genus $g$ is an embedded graph $T\subset S$ with a unique vertex such that all connected components of $S\setminus T$ are triangles (adjacent to exactly 3 edges of $T$). This paper gives formulas enumerating such triangulations (up to equivalence) on an oriented surface of given genus. {\em Une triangulation à un sommet} d'une surface orientée compacte $S$ de genre $g$ est un graphe $T\subset S$ qui a un unique sommet et dont toutes les faces (composantes connexes de $S\setminus T$) sont des triangles (incidentes à trois arêtes de $T$). Cet article donne des formules permettant d'énumérer ces triangulations.

math.CO↗
arXiv2001

Hooks and powers of parts in partitions

This paper shows that the number of hooks of length k contained in all partitions of n equals k times the number of parts of length k in all partitions of n. It contains also formulas for the moments (under uniform distribution) of k-th parts in partitions of n.

math.CO↗