A Comparison of Two Families of Algorithms for Symbolic Polynomials

Symbolic polynomials, whose exponents themselves are integer-valued multivariate polynomials, arise often in algorithm analysis. Unfortunately, modern computer algebra systems do not provide ample support for their algebraic structures. Basic operations involving symbolic polynomials are indeed trivial (addition, multiplication, derivatives); however, other crucial operations remain much more difficult, such as factorization and GCD. Watt has given two separate methods to solve these challenging problems. The first uses a change of variables, e.g. xn to X, xnm to Y. Doing so increases the number of variables potentially exponentially. Evaluation/Interpolation is used in the second algorithm. Projection methods introduce fewer variables but have larger degrees. We propose several adaptations to this idea to attempt to reduce the degree and number of required images. These include: sparse interpolation, evaluation point optimization and evaluating at primitive roots of unity. We give an in depth comparison of these algorithms, both empirically and theoretically

On systems of algebraic equations with parametric exponents

Kyushu University 21st Century COE Program Development of Dynamic Mathematics with High Functionality九州大学21世紀COEプログラム「機能数理学の構築と展開」International Conference on Symbolic and Algebraic Computation (ISSAC) 2004 : July 4-7, 2004 : University of Cantabria, Santander, SpainWe deal with systems of algebraic equations with parametric exponents. As the first step for solving such systems, we introduce a simple formulation and basic notions in ideal theory. Then we give a concrete method for most simple cases, univariate case and 0-dimensional case

On systems of algebraic equations with parametric exponents

We deal with systems of algebraic equations with parametric exponents. As the first step for solving such systems, we introduce a simple formulation and basic notions in ideal theory. Then we give a concrete method for most simple cases, univariate case and 0-dimensional case.Kyushu University 21st Century COE Program Development of Dynamic Mathematics with High Functionality九州大学21世紀COEプログラム「機能数理学の構築と展開」International Conference on Symbolic and Algebraic Computation (ISSAC) 2004 : July 4-7, 2004 : University of Cantabria, Santander, Spai