On the Inverse Problem Relative to Dynamics of the w Function

In this paper we shall study the inverse problem relative to dynamics of the w function which is a special arithmetic function and shall get some results.Comment: 11 page

On number fields with nontrivial subfields

What is the probability for a number field of composite degree $d$ to have a nontrivial subfield? As the reader might expect the answer heavily depends on the interpretation of probability. We show that if the fields are enumerated by the smallest height of their generators the probability is zero, at least if $d>6$. This is in contrast to what one expects when the fields are enumerated by the discriminant. The main result of this article is an estimate for the number of algebraic numbers of degree $d=e n$ and bounded height which generate a field that contains an unspecified subfield of degree $e$. If $n>\max\{e^2+e,10\}$ we get the correct asymptotics as the height tends to infinity

Lang's Conjecture and Sharp Height Estimates for the elliptic curves y2=x3+axy^{2}=x^{3}+ax

For elliptic curves given by the equation $E_{a}: y^{2}=x^{3}+ax$, we establish the best-possible version of Lang's conjecture on the lower bound of the canonical height of non-torsion points along with best-possible upper and lower bounds for the difference between the canonical and logarithmic height.Comment: published version. Lemmas 5.1 and 6.1 now precise (with resultant refinement to Theorem 1.2). Small corrections to

Anomalous Subvarieties—Structure Theorems and Applications

When a fixed algebraic variety in a multiplicative group variety is intersected with the union of all algebraic subgroups of fixed dimension, a key role is played by what we call the anomalous subvarieties. These arise when the algebraic variety meets translates of subgroups in sets larger than expected. We prove a Structure Theorem for the anomalous subvarieties, and we give some applications, emphasizing in particular the case of codimension two. We also state some related conjectures about the boundedness of absolute height on such intersections as well as their finitenes

The theory of the exponential differential equations of semiabelian varieties

The complete first order theories of the exponential differential equations of semiabelian varieties are given. It is shown that these theories also arises from an amalgamation-with-predimension construction in the style of Hrushovski. The theory includes necessary and sufficient conditions for a system of equations to have a solution. The necessary condition generalizes Ax's differential fields version of Schanuel's conjecture to semiabelian varieties. There is a purely algebraic corollary, the "Weak CIT" for semiabelian varieties, which concerns the intersections of algebraic subgroups with algebraic varieties.Comment: 53 pages; v3: Substantial changes, including a completely new introductio

Section Extension from Hyperbolic Geometry of Punctured Disk and Holomorphic Family of Flat Bundles

The construction of sections of bundles with prescribed jet values plays a fundamental role in problems of algebraic and complex geometry. When the jet values are prescribed on a positive dimensional subvariety, it is handled by theorems of Ohsawa-Takegoshi type which give extension of line bundle valued square-integrable top-degree holomorphic forms from the fiber at the origin of a family of complex manifolds over the open unit 1-disk when the curvature of the metric of line bundle is semipositive. We prove here an extension result when the curvature of the line bundle is only semipositive on each fiber with negativity on the total space assumed bounded from below and the connection of the metric locally bounded, if a square-integrable extension is known to be possible over a double point at the origin. It is a Hensel-lemma-type result analogous to Artin's application of the generalized implicit function theorem to the theory of obstruction in deformation theory. The motivation is the need in the abundance conjecture to construct pluricanonical sections from flatly twisted pluricanonical sections. We also give here a new approach to the original theorem of Ohsawa-Takegoshi by using the hyperbolic geometry of the punctured open unit 1-disk to reduce the original theorem of Ohsawa-Takegoshi to a simple application of the standard method of constructing holomorphic functions by solving the d-bar equation with cut-off functions and additional blowup weight functions

Sums and differences of four k-th powers

We prove an upper bound for the number of representations of a positive integer $N$ as the sum of four $k$-th powers of integers of size at most $B$, using a new version of the Determinant method developed by Heath-Brown, along with recent results by Salberger on the density of integral points on affine surfaces. More generally we consider representations by any integral diagonal form. The upper bound has the form $O_{N}(B^{c/\sqrt{k}})$, whereas earlier versions of the Determinant method would produce an exponent for $B$ of order $k^{-1/3}$ in this case. Furthermore, we prove that the number of representations of a positive integer $N$ as a sum of four $k$-th powers of non-negative integers is at most $O_{\epsilon}(N^{1/k+2/k^{3/2}+\epsilon})$ for $k \geq 3$, improving upon bounds by Wisdom.Comment: 18 pages. Mistake corrected in the statement of Theorem 1.2. To appear in Monatsh. Mat

On the ratio of consecutive gaps between primes

In the present work we prove a common generalization of Maynard-Tao's recent result about consecutive bounded gaps between primes and on the Erd\H{o}s-Rankin bound about large gaps between consecutive primes. The work answers in a strong form a 60 years old problem of Erd\"os, which asked whether the ratio of two consecutive primegaps can be infinitely often arbitrarily small, and arbitrarily large, respectively

A remark on the trace-map for the Silver mean sequence

In this work we study the Silver mean sequence based on substitution rules by means of a transfer-matrix approach. Using transfer-matrix method we find a recurrence relation for the traces of general transfer-matrices which characterizes electronic properties of the quasicrystal in question. We also find an invariant of the trace-map.Comment: 5 pages, minor improvements in style and presentation of calculation