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Evaluating Winding Numbers and Counting Complex Roots Through Cauchy Indices in Isabelle/HOL.

Wenda Li1, Lawrence C Paulson1

  • 1Computer Laboratory, University of Cambridge, Cambridge, UK.

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Summary

This study formalizes the winding number approximation using Cauchy indices in Isabelle theorem proving. This method, combined with the argument principle, aids in counting complex polynomial roots within specific domains.

Keywords:
Cauchy indexComputer algebraInteractive theorem provingIsabelle/HOLRoot countingThe Routh–Hurwitz stability criterionWinding number

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Area of Science:

  • Complex Analysis
  • Formal Verification
  • Computational Mathematics

Background:

  • The winding number quantifies path encirclements around a point in complex analysis.
  • The Cauchy index offers an approximation for winding number calculations.
  • Automated theorem proving can enhance complex mathematical formalizations.

Purpose of the Study:

  • To formalize the approximation of winding numbers using Cauchy indices within the Isabelle theorem prover.
  • To develop a computational tactic for evaluating winding numbers via Cauchy indices.
  • To apply these formalizations for counting complex polynomial roots in specified domains.

Main Methods:

  • Formalization of winding number and Cauchy index in Isabelle/HOL.
  • Development of a tactic to compute winding numbers using Cauchy indices.
  • Integration with the argument principle and remainder sequences for root counting.

Main Results:

  • A formal verification of the Cauchy index as an approximation for the winding number.
  • An implemented tactic in Isabelle for automated winding number evaluation.
  • Demonstrated application in counting complex roots of polynomials within rectangular and half-plane domains.

Conclusions:

  • The formalized Cauchy index approximation provides a practical tool for complex analysis within theorem provers.
  • This approach enhances the automated analysis of polynomial roots in computational mathematics.
  • The integration of formal methods with complex analysis opens new avenues for mathematical discovery.