Related papers: Affine Chabauty II

We extend the recent classification of Hilbert schemes with two Borel-fixed points to arbitrary characteristic. We accomplish this by synthesizing Reeves' algorithm for generating strongly stable ideals with the basic properties of…

The fractional Feynman-Kac equations describe the distribution of functionals of non-Brownian motion, or anomalous diffusion, including two types called the forward and backward fractional Feynman-Kac equations, where the fractional…

Interior Point Methods are widely used to solve Linear Programming problems. In this work, we present two primal affine scaling algorithms to achieve faster convergence in solving Linear Programming problems. In the first algorithm, we…

We prove some new degeneracy results for integral points and entire curves on surfaces; in particular, we provide the first example, to our knowledge, of a simply connected smooth variety whose sets of integral points are never…

The Chabauty--Kim method is a tool for finding the integral or rational points on varieties over number fields via certain transcendental $p$-adic analytic functions arising from certain Selmer schemes associated to the unipotent…

To solve the issue of segmenting rich texture images, a novel detection methods based on the affine invariable principle is proposed. Considering the similarity between the texture areas, we first take the affine transform to get numerous…

In this paper, we rigorously study an order 2 scheme that was previously proposed by some of the authors. A slight modification is proposed that enables us to prove the convergence of the scheme while simplifying in the same time the inner…

We prove uniform upper bounds on the number of integral points of bounded height on affine varieties. If $X$ is an irreducible affine variety of degree $d\geq 4$ in $\mathbb{A}^n$ which is not the preimage of a curve under a linear map…

We give an algorithm for finding the index of a positive outer automorphism of the free group, and prove the algorithm exits in a finite time.

In this PhD thesis we propose an algorithmic approach to the study of the Hilbert scheme. Developing algorithmic methods, we also obtain general results about Hilbert schemes. In Chapter 1 we discuss the equations defining the Hilbert…

We study the variation of the convergence Newton polygon of a differential equation along a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. Relying on work of the second author who investigated its…

We propose a robust classification algorithm for curves in 2D and 3D, under the special and full groups of affine transformations. To each plane or spatial curve we assign a plane signature curve. Curves, equivalent under an affine…

We extend the refined version of the Chabauty-Coleman bound on the number of rational points on a curve of genus g>1 to the case of bad reduction.

The central purpose of this article is to establish new inverse and implicit function theorems for differentiable maps with isolated critical points. One of the key ingredients is a discovery of the fact that differentiable maps with…

To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible…

We study the problem of efficiently constructing a curve C of genus 2 over a finite field F for which either the curve C itself or its Jacobian has a prescribed number N of F-rational points. In the case of the Jacobian, we show that any…

In this article we give the details of an effective point counting algorithm for genus two curves over finite fields of characteristic three. The algorithm has an application in the context of curve based cryptography. One distinguished…

We classify the irreducible components of the Hilbert scheme of $n$ points on non-reduced algebraic plane curves, and give a formula for the multiplicities of the irreducible components. The irreducible components are indexed by partitions…

Boundary problem for linear partial differential algebraic equations system with multiple characteristic curves is considered. It is supposed that matrix-functions pencil of the system under consideration is smoothly equivalent to special…

In this paper, we give two algorithms to compute preimages of curves under polynomial endomorphisms. In particular, this gives an efficient way of computing preimages of points. Furthermore, we explain the abstract setting under which one…