Diophantine perspectives to the exponential function and Euler’s factorial series
1University of Oulu Graduate School
2University of Oulu, Faculty of Science, Mathematics
|Online Access:||PDF Full Text (PDF, 1 MB)|
|Persistent link:|| http://urn.fi/urn:isbn:9789529418237
|Publish Date:|| 2019-04-30
|Thesis type:||Doctoral Dissertation
|Defence Note:||Academic dissertation to be presented for public discussion with the assent of the Doctoral Training Committee of Technology and Natural Sciences of the University of Oulu in Auditorium IT116, Linnanmaa, on 10 May 2019, at 12 noon.
Docent Tapani Matala-aho
Professor Michael Bennett
Professor Damien Roy
Professor Camilla Hollanti
The focus of this thesis is on two functions: the exponential function and Euler’s factorial series. By constructing explicit Padé approximations, we are able to improve lower bounds for linear forms in the values of these functions. In particular, the dependence on the height of the coefficients of the linear form will be sharpened in the lower bound.
The first chapter contains some necessary definitions and auxiliary results needed in later chapters.We give precise definitions for a transcendence measure and Padé approximations of the second type. Siegel’s lemma will be introduced as a fundamental tool in Diophantine approximation. A brief excursion to exterior algebras shows how they can be used to prove determinant expansion formulas. The reader will also be familiarised with valuations of number fields.
In Chapter 2, a new transcendence measure for e is proved using type II Hermite-Padé approximations to the exponential function. An improvement to the previous transcendence measures is achieved by estimating the common factors of the coefficients of the auxiliary polynomials.
The exponential function is the underlying topic of the third chapter as well. Now we study the common factors of the maximal minors of some large block matrices that appear when constructing Padé-type approximations to the exponential function. The factorisation of these minors is of interest both because of Bombieri and Vaaler’s improved version of Siegel’s lemma and because they are connected to finding explicit expressions for the approximation polynomials. In the beginning of Chapter 3, two general theorems concerning factors of Vandermonde-type block determinants are proved.
In the final chapter, we concentrate on Euler’s factorial series which has a positive radius of convergence in p-adic fields. We establish some non-vanishing results for a linear form in the values of Euler’s series at algebraic integer points. A lower bound for this linear form is derived as well.
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