Using a generalization due to Lerch [Bull. Int. Acad. François Joseph 3 (1896)] of a classical lemma of Zolotarev, employed in Zolotarev's proof of the law of quadratic reciprocity, we determine necessary and sufficient conditions for the difference of two Dedekind sums to be in 8ℤ. These yield new necessary conditions for equality of two Dedekind sums. In addition, we resolve a conjecture of Girstmair [arXiv:1501.00655].
On montre comment écrire de grandes familles, avec de hautes multiplicités, de cas d’égalité pour l’inégalité de Stothers-Mason (si sont des polynômes premiers entre eux, le nombre exact de racines du produit dépasse de le plus grand des degrés des composantes . On développera pour cela des techniques polynomiales itératives inspirées des décompositions de Dunford-Schwartz et de fonctions de Belyi. Des exemples d’application avec les conjectures ou de M. Hall sont développés.
We study some functional equations between Mahler measures of genus-one curves in terms of isogenies between the curves. These equations have the potential to establish relationships between Mahler measure and especial values of -functions. These notes are based on a talk that the author gave at the “Cuartas Jornadas de Teoría de Números”, Bilbao, 2011.
We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.
Let be a number field. It is well known that the set of recurrencesequences with entries in is closed under component-wise operations, and so it can be equipped with a ring structure. We try to understand the structure of this ring, in particular to understand which algebraic equations have a solution in the ring. For the case of cyclic equations a conjecture due to Pisot states the following: assume is a recurrence sequence and suppose that all the have a root in the field ; then (after...
We compute, in a unified way, the equations of all hyperelliptic modular curves. The main tool is provided by a class of modular functions introduced by Newman in 1957. The method uses the action of the hyperelliptic involution on the cusps.
Let be the family of all 2-connected plane triangulations with vertices of degree three or six. Grünbaum and Motzkin proved (in dual terms) that every graph P ∈ has a decomposition into factors P₀, P₁, P₂ (indexed by elements of the cyclic group Q = 0,1,2) such that every factor consists of two induced paths of the same length M(q), and K(q) - 1 induced cycles of the same length 2M(q). For q ∈ Q, we define an integer S⁺(q) such that the vector (K(q),M(q),S⁺(q)) determines the graph P (if P is...