相关论文: Vaaler's theorem in number fields
Darmon's conjecture on a relation between cyclotomic units over real quadratic fields and certain algebraic regulators was recently solved by Mazur and Rubin by using their theory of Kolyvagin systems. In this paper, we formulate a…
In a series of articles published in the C.R. Paris more than a century ago, T. P\'epin announced a list of ``theorems'' concerning the solvability of diophantine equations of the type $ax^4 + by^4 = z^2$. In this article, we show how to…
Diophantine approximation explores how well irrational numbers can be approximated by rationals, with foundational results by Dirichlet, Hurwitz, and Liouville culminating in Roth's theorem. Schmidt's subspace theorem extends Roth's results…
This paper creates a link between \textit{Tropical Geometry} and \textit{Difference Algebra}. The main result is a difference version of \textit{Kapranov's Theorem}. In this theorem, we extend Kapranov's Theorem to the case of a Laurent…
In this note we generalize and prove a recent conjecture of Varchenko concerning the number of critical points of a (multivalued) meromorphic function $\phi$ on an algebraic manifold. Under certain conditions, this number turns out to…
A vector variational principle is proved.
This paper presents a proof of Gallai's Theorem, adapted from A. Soifer's presentation in The Mathematical Coloring Book of E. Witt's 1952 proof of Gallai's Theorem.
We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.
Let $\alpha$ be an irrational real number. We show that the set of $\epsilon$-badly approximable numbers \[ \mathrm{Bad}^\varepsilon (\alpha) := \{x\in [0,1]\, : \, \liminf_{|q| \to \infty} |q| \cdot \| q\alpha -x \| \geq \varepsilon \} \]…
We prove an analogue of the prime number theorem for finite fields.
We consider the divergent fractional Laplace operator presented in [Dipierro-Savin-Valdinoci, Rev. Mat. Iberoam.] and we prove three types of results. Firstly, we show that any given function can be locally shadowed by a solution of a…
In this article we bound the number of remarkable values of a polynomial vector field. The proof is short and based on Jouanolou's theorem about rational first integrals of planar polynomial derivations. Our bound is given in term of the…
Noether's problem is classical and very important problem in algebra. It is an intrinsically interesting problem in invariant theory, but with far reaching applications in the sutdy of moduli spaces, PI-algebras, and the Inverse problem of…
In this article, we prove a conjecture of G\"unter K\"ohler on the ambiguity of the quadratic field in the definition of Hecke theta series by deriving it from a similar statement on two-dimensional Galois representations induced from…
The Subspace Theorem is a powerful tool in number theory. It has appeared in various forms and been adapted and improved over time. It's applications include diophantine approximation, results about integral points on algebraic curves and…
Recent results of Freitas, Kraus, Sengun and Siksek give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over various number fields. In this paper, we prove asymptotic results about the solutions of the Diophantine…
The use of Cauchy's method in proving the well-known Euler formula is an object of many controversies. The purpose of this paper is to prove that the Cauchy's method applies for convex polyhedra and not only for them, but also for surfaces…
The inhomogeneous Khintchine-Groshev Theorem is a classical generalization of Khintchine's Theorem in Diophantine approximation, by approximating points in $\mathbb{R}^m$ by systems of linear forms in $n$ variables. Analogous to the…
We prove a discrete time analogue of 1967 Moser's normal form of real analytic perturbations of vector fields possessing an invariant, reducible, Diophantine torus; in the case of diffeomorphisms too, the persistence of such an invariant…
In this exposition we discuss the theory of algebraic extensions of valued fields. Our approach is mostly through Galois theory. Most of the results are well-known, but some are new. No previous knowledge on the theory of valuations is…