Computer Algebra and Polynomials

Algebra and number theory have always been counted among the most beautiful mathematical areas with deep proofs and elegant results. However, for a long time they were not considered that important in view of the lack of real-life applications. This has dramatically changed: nowadays we find applications of algebra and number theory frequently in our daily life.

This book focuses on the theory and algorithms for polynomials over various coefficient domains such as a finite field or ring. The operations on polynomials in the focus are factorization, composition and decomposition, basis computation for modules, etc. Algorithms for such operations on polynomials have always been a central interest in computer algebra, as it combines formal (the variables) and algebraic or numeric (the coefficients) aspects.

The papers presented were selected from the Workshop on Computer Algebra and Polynomials, which was held in Linz at the Johann Radon Institute for Computational and Applied Mathematics (RICAM) during November 25-29, 2013, at the occasion of the Special Semester on Applications of Algebra and Number Theory.

Inhalt

An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications in Enumerative Combinatorics.- Moving Curve Ideals of Rational Plane Parametrizations.- Survey on Counting Special Types of Polynomials.- Orbit Closures of Linear Algebraic Groups.- Symbolic Solutions of First-Order Algebraic ODEs.- Ore Polynomials in Sage.- Giac and GeoGebra - Improved Gröbner Basis Computations.- Polar Varieties Revisited.- A Note on a Problem Proposed by Kim and Lisonek.- Fast Algorithms for Refined Parameterized Telescoping in Difference Fields.- Some Results on the Surjectivity of Surface Parametrizations.- Rational Normal Curves as Set-Theoretic Complete Intersections of Quadrics.