Geometric Function Atlas: certified computing for geometric function theory in Python

2026-09-07

AI Summary

本文介绍了一个开源Python包，用于解决几何函数理论中的极值问题，通过计算泰勒系数、Fekete-Szegő常数等，并提供不同级别的验证方法。

Abstract

We describe geometric-function-atlas, our open-source Python package for the sharp extremal problems of geometric function theory. We organise it around a catalogue of thirty-nine Ma--Minda starlike generators. From this catalogue we compute exact Taylor coefficients, closed-form Fekete--Szeg\H{o} constants, exact coefficients of the Ma--Minda extremal function, and admissibility screens. Our verifier answers membership questions for normalised polynomials at three levels of evidence: a floating-point grid screen, an exact sufficient condition decided in rational arithmetic, and a certified interval enclosure at the worst screened point. Every answer names the level at which we obtained it. We ship a checksummed artifact snapshot with three hundred and six coefficient certificates and seven hundred and two directed inclusion radii. Eight reviewed radius lanes carry certificates whose proof chains we replay symbolically, and we re-execute every coefficient certificate through our exact Schur-parameter machinery on request. We emit all results through one versioned envelope that records the method, the evidence status, the assumptions, and the artifact identifiers. Two optional laboratories apply the same discipline to cryptographic S-box metrics and to image-quality metrics. We present our design, state as propositions what each tier establishes, follow one radius lane from screen to replayed certificate, report measured timings, and place our package among symbolic-algebra, rigorous-numerics, and mathematical-database software. We release geometric-function-atlas under the MIT licence on the Python Package Index and at https://github.com/Prasanna28Devadiga/geometric-function-atlas.

Problem

• geometric function theory
• extremal problems
• Taylor coefficients
• Fekete-Szegő constants
• Ma-Minda starlike generators

Innovation

• geometric function theory
• exact computation
• certified interval enclosure
• open-source Python package
• Ma-Minda starlike generators