Related papers: Perfect Powers: Pillai's works and their developme…
In this note we give some remarks and improvements on a recent paper of us [3] about an optimization problem for the $p-$Laplace operator that were motivated by some discussion the authors had with Prof. Cianchi.
We consider Diophantine equations of the shape $ f(x) = g(y) $, where the polynomials $ f $ and $ g $ are elements of power sums. Using a finiteness criterion of Bilu and Tichy, we will prove that under suitable assumptions infinitely many…
Myasnikov, Ushakov, and Won introduced power circuits in 2012 to construct a polynomial-time algorithm for the word problem in the Baumslag group, which has a non-elementary Dehn function. Power circuits are computational structures that…
In this paper some open problems for Painlev\'e equations are discussed. In particular the following open problems are described: (i) the Painlev\'e equivalence problem; (ii) notation for solutions of the Painlev\'e equations; (iii)…
This text focuses on actions on 1-manifolds. We present a (non exhaustive) list of very concrete open questions in the field, each of which is discussed in some detail and complemented with a large list of references, so that a clear…
This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear subspaces generated by the best…
In this work we investigate two distinct extensions of the deformation procedure introduced in former works on deformed defects. The first extension deals with the use of deformation functions which can assume complex values, and the second…
Basic facts and definitions of conformal moduli of rings and quadrilaterals are recalled. Some computational methods are reviewed. For the case of quadrilaterals with polygonal sides, some recent results are given. Some numerical…
In this paper we study elliptic curves which have a number of points whose coordinates are in arithmetic progression. We first motivate this diophantine problem, prove some results, provide a number of interesting examples and, finally…
In this paper we consider the Diophantine equation \begin{align*}b^k +\left(a+b\right)^k &+ \cdots + \left(a\left(x-1\right) + b\right)^k=\\ &=d^l + \left(c+d\right)^l + \cdots + \left(c\left(y-1\right) + d\right)^l, \end{align*} where…
A Thesis about Euler discussing the possibilities and limits of his method of work in Mathematics.
A short account of recent existence and multiplicity theorems on the Dirichlet problem for an elliptic equation with $(p,q)$-Laplacian in a bounded domain is performed. Both eigenvalue problems and different types of perturbation terms are…
This survey article is an introduction to Diophantine Geometry at a basic undergraduate level. It focuses on Diophantine Equations and the qualitative description of their solutions rather than detailed proofs.
We place the theory of metric Diophantine approximation on manifolds into a broader context of studying Diophantine properties of points generic with respect to certain measures on $\Bbb R^n$. The correspondence between multidimensional…
We survey recent developments in the theory of achievement sets and present a substantial collection of open problems.
We give an induction-free axiom system for diophantine correct open induction. We relate the problem of whether a finitely generated ring of Puiseux polynomials is diophantine correct to a problem about the value-distribution of a tuple of…
In the first part, we revisit the theory of Drinfeld modular curves and $\pi$-adic Drinfeld modular forms for GL(2) from the perfectoid point of view. In the second part, we review open problems for families of Drinfeld modular forms for…
Particular solutions of the Benney equations are constructed. Their properties are discussed.
Boolean ultrapowers extend the classical ultrapower construction to work with ultrafilters on any complete Boolean algebra, rather than only on a power set algebra. When they are well-founded, the associated Boolean ultrapower embeddings…
We discuss the work of Abhyankar on dicritical divisors with a special focus on the algebraic aspects of this work. We also discuss related work on local quadratic transforms, infinitely near points and Rees valuation rings of an ideal.