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

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.