Related papers: Explicit Bounds for Linear Forms in the Exponentia…

Let $m\ge 2$ be an integer, $K$ an algebraic number field and $\alpha\in K\setminus \{0,-1\}$ with sufficiently small absolute value. In this article, we provide a new lower bound for linear form in…

Let $\mathbb{I}$ denote an imaginary quadratic field or the field $\mathbb{Q}$ of rational numbers and $\mathbb{Z}_{\mathbb{I}}$ its ring of intergers. We shall prove an explicit Baker type lower bound for $\mathbb{Z}_{\mathbb{I}}$-linear…

In this note, we shall give an improved lower bound for the argument of a power of a given algebraic number which has absolute value one but is not a root of unity.

Let $n \ge 2$ be an integer and $\alpha_1, \ldots, \alpha_n$ be non-zero algebraic numbers. Let $b_1, \ldots , b_n$ be integers with $b_n \not= 0$, and set $B = \max\{3, |b_1|, \ldots , |b_n|\}$. For $j =1, \ldots, n$, set $h^* (\alpha_j) =…

A sharp explicit estimate is proved for the difference $e^\beta-\alpha$ when $\alpha$ and $\beta$ are nonzero algebraic numbers.

We show that if $\lambda_1,\ldots,\lambda_k$ are algebraic numbers, then $$|A+\lambda_1\cdot A+\dots+\lambda_k\cdot A|\geq H(\lambda_1,\ldots,\lambda_k)|A|-o(|A|)$$ for all finite subsets $A$ of $\mathbb{C}$, where…

We obtain general criteria for giving a lower bound on the degree of numbers of the form $\prod_{n=1}^\infty \left(1+\frac{b_n}{\alpha_n}\right)$ or of the form $\prod_{m=1}^\infty \left(1+ \sum_{n=1}^\infty…

We provide a technique to obtain explicit bounds for problems that can be reduced to linear forms in three complex logarithms of algebraic numbers. This technique can produce bounds significantly better than general results on lower bounds…

Linear forms in logarithms have an important role in the theory of Diophantine equations. In this article, we prove explicit $p$-adic lower bounds for linear forms in $p$-adic logarithms of rational numbers using Pad\'e approximations of…

The aim of this manuscript is to derive bounds on the moduli of eigenvalues of special type of rational matrices of the form $T(\lambda) = \displaystyle -B_0 +I\lambda +\frac{B_1}{\lambda-\alpha_1}+ \dots+ \frac{B_m}{\lambda-\alpha_m}$,…

We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higher-order computation and linear algebra. This language extends the Lambda-calculus…

For $\lambda \in (1/2, 1)$ and $\alpha$, we consider sets of numbers $x$ such that for infinitely many $n$, $x$ is $2^{-\alpha n}$-close to some $\sum_{i=1}^n \omega_i \lambda^i$, where $\omega_i \in \{0,1\}$. These sets are in Falconer's…

A criterion is given for studying (explicit) Baker type lower bounds of linear forms in numbers $1,\Theta_1,...,\Theta_m\in\mathbb{C}^*$ over the ring $\mathbb{Z}_{\mathbb{I}}$ of an imaginary quadratic field $\mathbb{I}$. This work deals…

In this paper, we prove effective upper bounds for effective sections of line bundles on projective varieties and hermitian line bundles on arithmetic varieties in terms of the volumes. They are effective versions of the Hilbert--Samuel…

We give new bounds for $\sum_{{a, m ,n}}\alpha_{m}\beta_n\nu_a {\textrm e}\left(\frac{a\overline m}{n}\right)$ where $\alpha_{m}$, $\beta_n$ and $\nu_a$ are arbitrary coefficients, improving upon a result of Duke, Friedlander and Iwaniec…

We prove the Oppenheim conjecture for indefinite ternary diagonal forms of the type $x^{2}+y^{2} -\alpha z^{2}$ where $ \alpha $ is an irrational number. Our method is explicit in the sense that we are able to construct a solution to the…

The algebraic $\lambda$-calculus is an extension of the ordinary $\lambda$-calculus with linear combinations of terms. We establish that two ordinary $\lambda$-terms are equivalent in the algebraic $\lambda$-calculus iff they are…

This paper studies explicit and theoretical bounds for several interesting quantities in number theory, conditionally on the Generalized Riemann Hypothesis. Specifically, we improve the existing explicit bounds for the least quadratic…

This is a review of several results related to distribution of powers and combination of powers modulo 1. We include a proof that given a sequence of real numbers $\theta_n$, it is possible to get an $\alpha$ (given $\lambda \ne 0$), or a…

Let $\Lambda$ be an $n$-Auslander algebra with global dimension $n+1$. In this paper, we prove that $\Lambda$ is representation-finite if and only if the number of non-isomorphic indecomposable $\Lambda$-modules with projective dimension…