Related papers: Effective Methods for Norm-Form Equations
Exactly solvable variable parametric Burgers type equations in one-dimension are introduced, and two different approaches for solving the corresponding initial value problems are given. The first one is using the relationship between the…
In this paper we will give a unified proof of several results on the sovability of systems of certain equations over finite fields, which were recently obtained by Fourier analytic methods. Roughly speaking, we show that almost all systems…
We present two new algorithms for solving norm equations over global function fields with at least one infinite place of degree 1 and no wild ramification. The first of these is a substantial improvement of a method due to Ga\'{a}l and…
This paper applies the modular approach to obtain effectively computable bounds for Fermat-type equations over number fields, while also discussing the differences and obstructions that arise when considering such equations over totally…
Wiles' proof of Fermat's last theorem initiated a powerful new approach towards the resolution of certain Diophantine equations over $\mathbb{Q}$. Numerous novel obstacles arise when extending this approach to the resolution of Diophantine…
We implement the Unified Transform Method of Fokas as a numerical method to solve linear partial differential equations on the half-line. The method computes the solution at any x and t without spatial discretization or time stepping. With…
For many finite groups, the Inverse Galois Problem can be approached through modular/automorphic Galois representations. This is a report explaining the basic strategy, ideas and methods behind some recent results. It focusses mostly on the…
We apply inequalities from the theory of linear forms in logarithms to deduce effective results on S-integral points on certain higher-dimensional varieties when the cardinality of S is sufficiently small. These results may be viewed as a…
In 1975, Goldfeld gave an effective solution to Gauss's conjecture on the class numbers of imaginary quadratic fields. In this paper, we generalize Goldfeld's theorem to the setting of totally real number fields.
The problem to determine an explicit one-parameter power form representation of the proper Zolotarev polynomials of degree $n$ and with uniform norm $1$ on $[-1,1]$ can be traced back to P. L. Chebyshev. It turned out to be complicated,…
In this paper we study the set of rational solutions of equations defined by power sums symmetric polynomials with coefficients in a finite field. We do this by means of applying a methodology which relies on the study of the geometry of…
We discuss the following two problems: 1) The properties of the multiple zeta-values and their generalizations, multiple polylogarithms at N-th roots of unity; 2) The action of the absolute Galois group on the pro-l-completion of the…
In this paper, we solve certain Fermat-type partial differential-difference equations for finite order entire functions of several complex variables. These results are significant generalizations of some earlier findings, especially those…
This paper considers the problems of solving monotone variational inequalities with H\"older continuous Jacobians. By employing the knowledge of H\"older parameter $\nu$, we propose the $\nu$-regularized extra-Newton method within at most…
Using properties of Gauss and Jacobi sums, we derive explicit formulas for the number of solutions to a diagonal equation of the form $x_1^{2^m}+\dots+x_n^{2^m}=0$ over a finite field of characteristic $p\equiv\pm 3\pmod{8}$. All of the…
In this paper we give a unified approach in categorical setting to the problem of finding the Galois closure of a finite cover, which includes as special cases the familiar finite separable field extensions, finite unramified covers of a…
We study the module of universal norms associated with a de Rham $p$-adic Galois representation in a perfectoid field extension. In particular, we compute precisely this module when the Hodge-Tate weights of a representation are greater…
In this article we solve a general class of sextic equations. The solution follows if we consider the $j$-invariant and relate it with the polynomial equation's coefficients. The form of the solution is a relation of Rogers-Ramanujan…
One of the first parametrised Thue equations, $$\left| X^3 - (n-1)X^2 Y - (n+2) XY^2 - Y^3 \right| = 1,$$ over the integers was solved by E. Thomas in 1990. If we interpret this as a norm-form equation, we can write this as $$\left|…
We extend the Brauer-Siegel theorem to new families of number fields, both in the classical setting of asymptotically bad families and in the more general framework due to Tsfasman and Vl\u{a}du\c{t} of asymptotically exact families. We…