| Abstract: |
| Computing the Puiseux series of complex plane algebraic curves is a cornerstone of computational algebraic geometry, with applications ranging from robotics to theoretical physics. While the Newton-Puiseux algorithm provides a rigorous framework for desingularization, its practical efficiency is often limited by the prohibitive cost of exact arithmetic over algebraic number fields. Conversely, purely numerical approaches struggle with the inherent instability of singularities, where infinitesimal perturbations can collapse the branching structure of the curve.
This work proposes a hybrid symbolic-numeric strategy that leverages validated numerics to compute rigorous interval representations of Puiseux coefficients while bypassing the cost of computing with exact algebraic numbers. Our approach utilizes the modular-numeric strategy introduced in Poteaux`s thesis, where the structure of the Puiseux series is first inferred from an execution of Newton-Puiseux over a finite field, by choosing an appropriate prime number. This structural data is encoded into a polygon tree, which is then used as a blueprint guiding the numerical phase.
To bridge the symbolic-numeric gap, we develop routines called filters to correctly associate the recursive calls of the numeric algorithm with the branches of these trees, thus ensuring the preservation of Newton polygons and multiplicities at every step. By doing so, we achieve a certified result where traditional interval methods alone would otherwise fail to capture the structure
of the singularity. This work demonstrates how the synergy between computer algebra and validated numerics can resolve highly sensitive singular problems with both speed and mathematical rigor. |
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