Counting rational points on a certain exponential-algebraic surface

Jonathan Pila

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Displaying similar documents to “Counting rational points on a certain exponential-algebraic surface”

Complete solutions of a family of cubic Thue equations

Alain Togbé (2006)

Journal de Théorie des Nombres de Bordeaux

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In this paper, we use Baker’s method, based on linear forms of logarithms, to solve a family of Thue equations associated with a family of number fields of degree 3. We obtain all solutions to the Thue equation $\begin{matrix} Φ_{n} (x, y) & = x^{3} + (n^{8} + 2 n^{6} - 3 n^{5} + 3 n^{4} - 4 n^{3} + 5 n^{2} - 3 n + 3) x^{2} y \\ - (n^{3} - 2) n^{2} x y^{2} - y^{3} = \pm 1, \end{matrix}$ for $n \geq 0$ .

Logarithmic Surfaces and Hyperbolicity

Gerd Dethloff, Steven S.-Y. Lu (2007)

Annales de l’institut Fourier

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In 1981 J. Noguchi proved that in a logarithmic algebraic manifold, having logarithmic irregularity strictly bigger than its dimension, any entire curve is algebraically degenerate. In the present paper we are interested in the case of manifolds having logarithmic irregularity equal to its dimension. We restrict our attention to Brody curves, for which we resolve the problem completely in dimension 2: in a logarithmic surface with logarithmic irregularity...

Non-embeddability of general unipotent diffeomorphisms up to formal conjugacy

Javier Ribón (2009)

Annales de l’institut Fourier

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The formal class of a germ of diffeomorphism $ϕ$ is embeddable in a flow if $ϕ$ is formally conjugated to the exponential of a germ of vector field. We prove that there are complex analytic unipotent germs of diffeomorphisms at $ℂ^{n}$ ( $n > 1$ ) whose formal class is non-embeddable. The examples are inside a family in which the non-embeddability is of geometrical type. The proof relies on the properties of some linear functional operators that we obtain through the study of polynomial families of diffeomorphisms...

Lower bounds for a certain class of error functions

J. Herzog, P. R. Smith (1992)

Acta Arithmetica

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Equidistribution of Small Points, Rational Dynamics, and Potential Theory

Matthew H. Baker, Robert Rumely (2006)

Annales de l’institut Fourier

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Given a rational function $ϕ (T)$ on $ℙ^{1}$ of degree at least 2 with coefficients in a number field $k$ , we show that for each place $v$ of $k$ , there is a unique probability measure $μ_{ϕ, v}$ on the Berkovich space $ℙ_{Berk, v}^{1} / ℂ_{v}$ such that if ${z_{n}}$ is a sequence of points in $ℙ^{1} (\bar{k})$ whose $ϕ$ -canonical heights tend to zero, then the $z_{n}$ ’s and their $Gal (\bar{k} / k)$ -conjugates are equidistributed with respect to $μ_{ϕ, v}$ . The proof uses a polynomial lift $F (x, y) = (F_{1} (x, y), F_{2} (x, y))$ of $ϕ$ to construct a two-variable Arakelov-Green’s function $g_{ϕ, v} (x, y)$ for each $v$ . The measure $μ_{ϕ, v}$ is...

On the mean values of an analytic function.

G. S. Srivastava, Sunita Rani (1992)

Annales Polonici Mathematici

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Let f(z), $z = r e^{i θ}$ , be analytic in the finite disc |z| < R. The growth properties of f(z) are studied using the mean values $I_{δ} (r)$ and the iterated mean values $N_{δ, k} (r)$ of f(z). A convexity result for the above mean values is obtained and their relative growth is studied using the order and type of f(z).