What does this research mean for the field?

Integrating an invertible infinitesimal unit within the field of formal Puiseux series and utilizing a 'Zeta axis' geometric visualization provides a rigorous mathematical framework for resolving indeterminate forms and extracting exact topological data in complex analysis. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To construct an algebraic and geometric foundation for complex analysis that resolves indeterminate forms using infinitesimal series and visualizes complex functions through invariant-based 3D surfaces.

August 29, 2026Open Access

The Zeta Axis II: A Geometric Framework for Complex Analysis via Formal Puiseux Series and Exact Certification

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Authors

JRJorge Vicente Romero

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Implication

Mathematical study reveals harmonic geometric structures in rational maps via formal Puiseux series, highlighting an exact certification framework that avoids computational resultants.

Key Points

To construct an algebraic and geometric foundation for complex analysis that resolves indeterminate forms using infinitesimal series and visualizes complex functions through invariant-based 3D surfaces.
Substituted classical zero with an invertible infinitesimal unit in the field of formal Puiseux series.
Introduced a third coordinate axis representing differential invariants classified by the five connected algebraic subgroups of the projective general linear group PGL_2(C).
Applied local-to-global topological identities across eight algebraic groups to compute divisors, exponents, and monodromy using exact orders and winding numbers.
Transformed indeterminate forms into determinate values and converted poles into elements with negative algebraic order.
Demonstrated that Gaussian and mean curvature measures categorize surface topology into folds, saddles, and flat points that directly match rational map poles, zeros, and critical points.
Achieved exact numerical certification of tangent-cone monodromy and branch data without using resultant calculations or standard sampling approximations.

Cite This Study

Jorge Vicente Romero (2026) studied this question.

synapsesocial.com/papers/6a9299db8e5d7d1fc0c120f2 https://doi.org/10.5281/zenodo.22135029

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Implication

Mathematical study reveals harmonic geometric structures in rational maps via formal Puiseux series, highlighting an exact certification framework that avoids computational resultants.

The Zeta Axis II: A Geometric Framework for Complex Analysis via Formal Puiseux Series and Exact Certification

Discussion

Member takes

Implication

Key Points

Cite This Study

The Zeta Axis II: A Geometric Framework for Complex Analysis via Formal Puiseux Series and Exact Certification

Discussion

Member takes

Implication

Key Points

Cite This Study

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