Physical Sciences and Mathematics Commons^™

Solving Polynomial Systems Via Triangular Decomposition, Changbo Chen

Electronic Thesis and Dissertation Repository

Finding the solutions of a polynomial system is a fundamental problem with numerous applications in both the academic and industrial world. In this thesis, we target on computing symbolically both the real and the complex solutions of nonlinear polynomial systems with or without parameters. To this end, we improve existing algorithms for computing triangular decompositions. Based on that, we develop various new tools for solving polynomial systems and illustrate their effectiveness by applications.

We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a weakened notion of a …

Algorithmic Contributions To The Theory Of Regular Chains, Wei Pan

Electronic Thesis and Dissertation Repository

Regular chains, introduced about twenty years ago, have emerged as one of the major

tools for solving polynomial systems symbolically. In this thesis, we focus on different

algorithmic aspects of the theory of regular chains, from theoretical questions to high-

performance implementation issues.

The inclusion test for saturated ideals is a fundamental problem in this theory.

By studying the primitivity of regular chains, we show that a regular chain generates

its saturated ideal if and only if it is primitive. As a result, a family of inclusion tests

can be detected very efficiently.

The algorithm to compute the regular GCDs …