Algebrai számokkal kapcsolatos algoritmusok és számítások

Abstract

The dissertation is about algorithms of algebraic numbers and related theoretical questions. An important aspect of the discussed algorithms is that they calculate exactly, not numerically. In this way we avoid the usual problem of numerical calculations, the rounding errors, but the exact algorithms are usually slower, and they often suffer from coefficient explosion. Avoiding this problem and maintaining the running time (i.e. ensuring polynomial time) is an important aspect of exact algorithms. The dissertation has two major parts. The first half deals with the computational complexity of operations and algorithms involving algebraic numbers. We calculate the running time of the main operations in algebraic number fields (using some existing results). We also analyse some special operations like comparison between real algebraic numbers and integer rounding. These results enable us to analyse two algorithms, the Bareiss algorithm (a generalisation of the Gaussian elimination) and the LLL algorithm (a lattice reduction algorithm) on algebraic numbers using exact arithmetic, and analyse their complexity. Their behaviour is well-known for integers, but we extend them to algebraic integers, and prove that their running time remains polynomial. The second half of the dissertation is about special algebraic numbers: roots of integer polynomials whose all roots are outside of the complex unit circle. Such polynomials are called expansive polinomials. We examine the question how to decide whether a polynomial with integer coefficients is expansive (without calculating its roots). There are some existing methods for this, but we introduce a new method that is guaranteed to avoid coefficient explosion. Then we examine how close the roots of an expansive integer polynomial can be to the unit circle, and give lower bounds on this distance in terms of the parameters of the polynomial. Then we construct a family of expansive polynomials that have roots very close to the unit circle, thus showing that the lower bounds are almost sharp.