SymPy: Master’s Thesis¶

Contents:

• Introduction
  • Symbolic manipulation systems
  • The pure Python approach
  • The author’s role
  • The current version
  • Origin of module’s name
  • Historical background
  • Presentations of this work
  • The structure of this thesis
  • A word on time measurement
  • Acknowledgements
• Notes on the internal implementation
  • Physical structure of the module
  • Logical structure of the module
    • Motivation
    • The zeroth level: L0
    • The first level: L1
    • The second level: L2
    • The third level: L3
    • Multiple–levels in practice
  • Polynomial representations
    • Dense polynomial representation
    • Sparse polynomial representation
    • Benchmarking polynomial representations
  • Categories, domains and ground types
    • Benchmarking ground types
  • Using Cython internally
    • Pure mode Cython in SymPy
    • Benchmarking pure mode Cython
  • Conclusions
• Algorithms for algebraic computations
  • Arithmetics of polynomials
  • Evaluation of polynomials
  • The Greatest Common Divisor
  • Square–free decomposition
  • Factorization of polynomials
    • Finite fields
    • Integers and rationals
    • Algebraic number fields
  • Gröbner bases
  • Root isolation
    • Real roots
    • Complex roots
  • Conclusions
• Gröbner bases and their applications
  • Short introduction to Gröbner bases
  • Construction of Gröbner bases
    • The notion of s–polynomials
    • What is a Gröbner basis?
    • Reduced Gröbner bases
    • Toy Buchberger algorithm
    • Termination of the algorithm
  • Computing Gröbner bases with SymPy
  • Admissible orderings of monomials
  • Specialization of Gröbner bases
    • Special case 1: Gauss’ algorithm
    • Special case 2: Euclid’s algorithm
  • Applications of Gröbner bases
    • Solving systems of polynomial equations
    • Algebraic relations in invariant theory
    • Integer optimization
    • Coloring of graphs
    • Algebraic geometry
    • Other applications
  • Complexity of computing Gröbner bases
  • Conclusions
• Final words and conclusions
  • Future plans
    • Polynomial arithmetics
    • Power series expansion
    • Partial fraction decomposition
    • Simplification of expressions
    • Cylindrical algebraic decomposition
    • Multiple algebraic extensions
    • Using modular techniques elsewhere
• Glossary
• Literature