Gusakova, Anna: Application of Probability Methods in Number Theory and Integral Geometry. 2018
Inhalt
- Introduction
- Counting Complex Algebraic Numbers on the Unit Circle
- General Method
- Connection of the Distribution of Algebraic Numbers and Zeroes of Random Polynomials
- Main Result
- Corollaries
- Proof of Theorem 2.3.2
- Proofs of Corollaries
- Counting Points with Algebraic Conjugate Coordinates
- Introduction
- Rectangles of Small Measure
- Some Technical Lemmas
- Proof of Theorem 3.2.1: Lower Bound
- Proof of Theorem 3.2.2: Lower Bound
- Proof of Theorem 3.2.3: Upper Bound
- Neighborhood of Curves
- Distribution of Algebraic Integers and Points with Conjugate Algebraic Integer Coordinates
- Affine Transformation of Random Simplices and Integral Geometry
- Main Result
- Random Points in Ellipsoids
- Integral Geometry Formulas
- Proofs: Part I
- Proof of Proposition 4.1.3
- Proof of Theorem 4.1.2
- Proof of Corollary 4.1.5
- Proofs of Theorem 4.2.1 and Theorem 4.2.2
- Proof of Corollary 4.2.1.1
- Proofs: Part II
- Some Results From Number Theory and Geometry of Numbers
- Random polynomials
- Integral Geometry
- Bibliography