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u-generation: solving systems of polynomials equation-by-equation

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Abstract

We develop a new method that improves the efficiency of equation-by-equation homotopy continuation methods for solving polynomial systems. Our method is based on a novel geometric construction and reduces the total number of homotopy paths that must be numerically continued. These improvements may be applied to the basic algorithms of numerical algebraic geometry in the settings of both projective and multiprojective varieties. Our computational experiments demonstrate significant savings obtained on several benchmark systems. We also present an extended case study on maximum likelihood estimation for rank-constrained symmetric \(n\times n\) matrices, in which multiprojective u-generation allows us to complete the list of ML degrees for \(n\le 6.\)

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Notes

  1. In this paper, we shall gloss over many details of how homotopy tracking is accomplished in practice and issues arising from the need to use approximations of points. For this, we refer the reader to introductory chapters of [31].

  2. Here, we use the fact that \(\pi \) is an open map, which follows from [27, 3.10].

  3. We use “general” throughout this article to mean “avoiding some proper Zariski closed set.” Here, this exceptional set has a simple description: \(\mathbb {V}(L)\) needs to be outside the set of subspaces in the Grassmannian \(\textrm{Gr}(n-c,n)\) that do not intersect X regularly. However, the randomized methods of homotopy continuation do not rely on knowing the exceptional set description: we use “general” without aiming to provide such a description later on (e.g., in Remark 2.5).

  4. The constant \(\gamma \) needs to avoid a real hypersurface (in fact, a union lines through the origin) in \(\mathbb {R}^2\cong \mathbb {C}\) containing “bad” choices: a choice of \(\gamma \) is “bad” if the homotopy paths of (3) cross.

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Funding

T.D. acknowledges support from an NSF Mathematical Sciences Postdoctoral Research Fellowship (DMS-2103310). Research of A.L. is supported in part by NSF DMS-2001267. Research of J.I.R. is supported by the Office of the Vice Chancellor for Research and Graduate Education at U.W. Madison with funding from the Wisconsin Alumni Research Foundation.

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Authors and Affiliations

  1. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195–4350, USA

    Timothy Duff

  2. School of Mathematics, Georgia Institute of Technology, 686 Cherry Street NW, Atlanta, GA, 30308, USA

    Anton Leykin

  3. Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI, 30308, USA

    Jose Israel Rodriguez

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  2. Anton Leykin
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All authors contributed to the preparation of the manuscript and the research it contains.

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Correspondence to Timothy Duff.

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All authors thank the Institute for Mathematics and its Applications in Minneapolis for hosting them for a series of “SageMath and Macaulay2” meetings in 2019-2020 where the ideas of this work were born.

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Timothy Duff: https://timduff35.github.io/timduff35/

Anton Leykin: https://antonleykin.math.gatech.edu/index.html

Jose Israel Rodriguez: https://sites.google.com/wisc.edu/jose/.

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Duff, T., Leykin, A. & Rodriguez, J.I. u-generation: solving systems of polynomials equation-by-equation. Numer Algor 95, 813–838 (2024). https://doi.org/10.1007/s11075-023-01590-1

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