Computer assistance for ``discovering'' formulas in system engineering and operator theory

J. William Helton - Mark Stankus

• Contents
• Strategies and Applications
  • Introduction
    • Background on polynomial equations and Ideals
      • A few polynomial equations
      • Solutions of Polynomial Equations And Ideals
    • A highly idealized picture
      • Basic ``idealized'' operations
      • Idealized strategies
      • New derivations of classical theorems: idealized picture
    • A practical picture
      • NCProcess commands
      • New derivations of classical theorems: practical picture
    • Algorithms and their properties
      • Obtaining smaller Bases; shrinking
      • Noncommutative elimination theory
    • A practical notation
    • Computational concerns
    • Acknowledgements
  • Prestrategy
    • Elimination
    • NCProcess
      • The input and output of NCProcess
      • A simple example of NCProcess1
      • Categories
      • Comparison of NCProcess and Categorize
    • Prestrategy
      • When to stop; the end game
  • Example: The Bart-Gohberg-Kaashoek-Van Dooren Theorem
    • Background
    • The problem
    • Solution via a prestrategy
    • The end game
    • Concluding Remarks
  • Examples: Matrix Completion Problems
    • The partially prescribed inverse problem
    • Unitary case of Parrot's Lemma
  • Strategies and motivated unknowns
    • Strategy
    • Computing decompositions
      • NCCollectOnVariables
      • Introducing a new motivated unknown
  • Example: Solving the Control Problem
    • Problem statement
    • Solving (HGRAIL) using NCProcess
      • Step 1 : Process and Collect
      • Step 2: The user attacks
      • Step 3
      • Step 4
    • End game
  • Monomial orders
  • Summary
  • Appendix to Part I: More details on NCCollectOnVariables
    • Collecting against a set of expressions
    • NCCollectOnVariables
• Theory and more details
  • Background on Ideals and Gröbner Bases
    • The Reduction Process
    • The Basis Algorithm
  • A Gröbner Basis Theorem on Elimination Ideals
  • Finding a small generating set for an ideal
    • The need to consider small generating sets
    • Preview of operations
    • The SmallBasis Operation
      • Approximation of the SmallBasis operation
    • The ShrinkBasis Operation
    • Gröbner Graph: Background for the RemoveRedundant Operation
      • Gröbner Basis Background
      • The graph
      • Definition of a Gröbner Graph
    • The RemoveRedundant operation
    • Respect for categories
      • Idealized shrinking by category
      • Practical shrinking by category
    • Remove Redundant Protected
  • Speeding up runs
    • DegreeCap and DegreeSumCap
    • Monomial Orders vs. Run time
  • Details of NCProcess
    • NCProcess1 command
    • NCProcess2 command
  • Adjoints
  • Example of discovering
  • Formal descriptions of pure lex and multigraded lex
    • An ordering on
    • Formal description of pure lex
    • Formal description of multigraded lex
  • Engineering motivation for the Control Problem (not for publication)
    • System definitions
    • Energy balance equations
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