Sparse Tensors and Subdivision Methods for Finding the Zero Set of Polynomial Equations
<|reference_start|>Sparse Tensors and Subdivision Methods for Finding the Zero Set of Polynomial Equations: Finding the solutions to a system of multivariate polynomial equations is a fundamental problem in mathematics and computer science. It involves evaluating the polynomials at many points, often chosen from a grid. In most current methods, such as subdivision, homotopy continuation, or marching cube algorithms, polynomial evaluation is treated as a black box, repeating the process for each point. We propose a new approach that partially evaluates the polynomials, allowing us to efficiently reuse computations across multiple points in a grid. Our method leverages the Compressed Sparse Fiber data structure to efficiently store and process subsets of grid points. We integrated our amortized evaluation scheme into a subdivision algorithm. Experimental results show that our approach is efficient in practice. Notably, our software \texttt{voxelize} can successfully enclose curves defined by two trivariate polynomial equations of degree $100$, a problem that was previously intractable.<|reference_end|>
@article{moroz2024sparse,
    title={Sparse Tensors and Subdivision Methods for Finding the Zero Set of
  Polynomial Equations},
    author={Guillaume Moroz (GAMBLE)},
    journal={Computer Algebra in Scientific Computing, Sep 2024, Rennes, France},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.09857},
    primaryClass={cs.CG cs.NA cs.SC math.NA}
}