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Numerical Nonlinear Algebra

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arxiv 2302.08585 v2 pith:NEIA6MYX submitted 2023-02-16 math.AG cs.NAmath.NA

Daniel J. Bates , Paul Breiding , Tianran Chen , Jonathan D. Hauenstein , Anton Leykin , Frank Sottile This is my paper · ORCID
classification math.AGcs.NAmath.NA
keywords numericalalgebraapplicationsequationsnonlinearalgebraicgeometrypolynomial
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Numerical nonlinear algebra is a computational paradigm that uses numerical analysis to study polynomial equations. Its origins were methods to solve systems of polynomial equations based on the classical theorem of B\'ezout. This was decisively linked to modern developments in algebraic geometry by the polyhedral homotopy algorithm of Huber and Sturmfels, which exploits the combinatorial structure of the equations and led to efficient software for solving polynomial equations. Subsequent growth of numerical nonlinear algebra continues to be informed by algebraic geometry and its applications. These include new approaches to solving, algorithms for studying positive-dimensional varieties, certification, and a range of applications both within mathematics and from other disciplines. With new implementations, numerical nonlinear algebra is now a fundamental computational tool for algebraic geometry and its applications. We survey some of these innovations and some recent applications.

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Cited by 3 Pith papers

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  1. The Maximum Likelihood Degree of Toric Models is Monotonic

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    The ML degree is monotone with respect to faces of the defining polytope for scaled toric models, settling a conjecture of Coons and Sullivant.

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    The (1,1,1,5) and (1,1,1,6) D2-D2-D2-D6 BPS systems yield 2032 and 5616 vacua, matching U-duality, while the analogous non-BPS system has no zero-energy vacua and six doubly-degenerate low-energy minima.

  3. Numerically Computing Galois Groups of Minimal Problems

    cs.CV 2025-07 conditional novelty 4.0 of 10

    A tutorial arguing that the Galois group and the Galois-width invariant characterize the intrinsic algebraic difficulty of minimal problems, with numerical monodromy code for the five-point problem.

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