pith. sign in

arxiv: 1907.00096 · v1 · pith:V2WUZ24Xnew · submitted 2019-06-28 · 💻 cs.MS · cs.DC· cs.SC· math.AG

Solving Polynomial Systems with phcpy

Jasmine Otto , Angus Forbes , Jan Verschelde This is my paper

Pith reviewed 2026-05-25 13:13 UTC · model grok-4.3

classification 💻 cs.MS cs.DCcs.SCmath.AG
keywords phcpyPHCpackpolynomial systemshomotopy continuationJupyter notebookGPU parallelizationnumerical solvingSTEM applications
0
0 comments X

The pith

phcpy now runs online in Jupyter with Python and SageMath kernels while adding GPU parallelization for larger polynomial systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents updates to phcpy, the Python scripting interface to the PHCpack package that solves systems of polynomials through homotopy continuation. It emphasizes that phcpy is now hosted on a JupyterHub server supporting Python 2, Python 3, and SageMath, so small systems solve in real time inside notebooks for interactive work. GPU parallelization is also added to increase speed and solution quality on bigger systems. The authors note that certain classes of polynomial systems common in mechanical design and nonlinear biological dynamics are well matched to these capabilities. A reader would care because the changes lower the barrier to reliable numerical solutions without requiring local installation or custom parallel code.

Core claim

The paper establishes that phcpy has been extended with JupyterHub availability and GPU support, allowing real-time interactive solution of small polynomial systems in notebooks and improved handling of much larger systems arising in STEM model design and analysis.

What carries the argument

phcpy as the scripting interface that exposes PHCpack's homotopy continuation methods, now with added online notebook access and GPU parallelization.

If this is right

  • Small polynomial systems can be explored interactively inside Jupyter notebooks without local setup.
  • Larger systems obtain solutions faster and with higher quality through GPU parallelization.
  • Model design problems in mechanical systems and phase-space analyses in biology can use these features directly.
  • Certain frequently arising polynomial system classes in STEM become more accessible for numerical solution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Integration with SageMath kernels may allow mixed symbolic-numeric workflows that combine phcpy with other computer-algebra tools.
  • Real-time notebook solving could support iterative prototyping loops in engineering design optimization.
  • The online availability might enable classroom or collaborative use cases where students or teams share solution sessions without installing software.

Load-bearing premise

The claimed real-time notebook performance and GPU speedups apply to the classes of polynomial systems that arise in the mechanical and biological applications mentioned.

What would settle it

A timing test on the JupyterHub server showing small systems take more than a few seconds to solve, or a benchmark on a large system showing no measurable improvement in speed or number of solutions when GPU parallelization is enabled.

Figures

Figures reproduced from arXiv: 1907.00096 by Angus Forbes, Jan Verschelde, Jasmine Otto.

Figure 1
Figure 1. Figure 1: The code snippet for the blackbox solver. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Tangent circles calculated by phcpy in response to user reparameterization of the system. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Interpolated elapsed performances. Precision is a crude measure of quality. Another motivation for quality up by parallelism is to compen￾sate for the cost overhead caused by arithmetic with power series. Power series are hybrid symbolic-numeric representations for algebraic curves. 3.3 Positive Dimensional Solution Sets Solving a system has evolved in meaning, from computing approximations of all its isol… view at source ↗
Figure 4
Figure 4. Figure 4: illustrates a reproduction of one synthesis result in the mechanism design literature [MW90]. Given five points, the problem is to determine the length of two bars so their coupler curve passes through the five given points [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
read the original abstract

The solutions of a system of polynomials in several variables are often needed, e.g.: in the design of mechanical systems, and in phase-space analyses of nonlinear biological dynamics. Reliable, accurate, and comprehensive numerical solutions are available through PHCpack, a FOSS package for solving polynomial systems with homotopy continuation. This paper explores new developments in phcpy, a scripting interface for PHCpack, over the past five years. For instance, phcpy is now available online through a JupyterHub server featuring Python2, Python3, and SageMath kernels. As small systems are solved in real-time by phcpy, they are suitable for interactive exploration through the notebook interface. Meanwhile, phcpy supports GPU parallelization, improving the speed and quality of solutions to much larger polynomial systems. From various model design and analysis problems in STEM, certain classes of polynomial system frequently arise, to which phcpy is well-suited.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript describes recent developments in phcpy, a Python scripting interface to the PHCpack package for solving systems of polynomial equations via homotopy continuation. It emphasizes online availability via a JupyterHub server supporting Python 2/3 and SageMath kernels for interactive real-time solving of small systems, GPU parallelization for larger systems, and suitability for polynomial systems arising in STEM applications such as mechanical design and biological dynamics.

Significance. If the described features function as stated, the work lowers barriers to using a reliable numerical solver for polynomial systems, enabling interactive exploration in notebooks and potentially scaling to larger problems via GPU support. This could benefit education and research in applied algebra and related fields, though the absence of any performance data or examples limits evaluation of practical gains over existing PHCpack usage.

major comments (2)
  1. [Abstract] Abstract and introduction: the claim that GPU parallelization 'improving the speed and quality of solutions to much larger polynomial systems' is presented without any supporting benchmarks, timings, scaling results, quality metrics, or even a single example computation. This assertion is load-bearing for the paper's positioning of phcpy as suitable for larger STEM problems.
  2. [Abstract] Abstract: the statement that 'small systems are solved in real-time by phcpy' and are 'suitable for interactive exploration' is asserted without reported timings, system sizes, or notebook usage examples to substantiate real-time performance on the JupyterHub deployment.
minor comments (1)
  1. The manuscript would benefit from a dedicated section or table listing the new phcpy functions or API changes introduced in the past five years, with references to the corresponding PHCpack routines.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed comments. We address each major point below and indicate planned revisions to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract and introduction: the claim that GPU parallelization 'improving the speed and quality of solutions to much larger polynomial systems' is presented without any supporting benchmarks, timings, scaling results, quality metrics, or even a single example computation. This assertion is load-bearing for the paper's positioning of phcpy as suitable for larger STEM problems.

    Authors: We agree that the manuscript presents the GPU parallelization feature without any benchmarks, timings, scaling results, quality metrics, or example computations. The paper describes the addition of this capability but does not evaluate its performance impact. We will revise the abstract and introduction to remove the unsubstantiated claim regarding improvements to speed and quality for much larger systems. revision: yes

  2. Referee: [Abstract] Abstract: the statement that 'small systems are solved in real-time by phcpy' and are 'suitable for interactive exploration' is asserted without reported timings, system sizes, or notebook usage examples to substantiate real-time performance on the JupyterHub deployment.

    Authors: We agree that the claims of real-time solving for small systems and suitability for interactive exploration lack supporting timings, system sizes, or usage examples. The manuscript notes the JupyterHub deployment with multiple kernels but provides no performance data. We will revise the abstract to remove or qualify these specific assertions. revision: yes

Circularity Check

0 steps flagged

No circularity; purely descriptive software announcement with no derivations or fitted predictions

full rationale

The paper is a software feature description for phcpy (a scripting interface to PHCpack). It states availability via JupyterHub and GPU support but contains no equations, no parameter fitting, no predictions of quantities from inputs, and no derivation chain. The GPU claim is an unsupported assertion rather than a reduction of a result to its own inputs. No self-citations, ansatzes, or uniqueness theorems appear in the provided text. This matches the default expectation of no circularity for non-derivational papers.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are introduced; the paper is a description of software features.

pith-pipeline@v0.9.0 · 5685 in / 978 out tokens · 17969 ms · 2026-05-25T13:13:19.729773+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

  1. [1]

    Bartzos, I

    [BELT18] E. Bartzos, I. Z. Emiris, J. Legersky, and E. Tsigaridas. On the maximal number of real embeddings of spatial minimally rigid graphs . In the Proceedings of the 2018 International Symposium on Symbolic and Algebraic Computation (ISSAC 2018), pages 55-62, ACM

    work page 2018

  2. [2]

    [Bertini2.0] Bertini 2.0: The redevelopment of Bertini in C++

    DOI 10.1145/3208976.3208994. [Bertini2.0] Bertini 2.0: The redevelopment of Bertini in C++. https://github.com/bertiniteam/b2 [BF91] J. Backelin and R. Frberg. How we proved that there are exactly 924 cyclic 7-roots. In the Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation (ISSAC’91), pages 103-111, ACM,

    work page doi:10.1145/3208976.3208994 1991

  3. [3]

    divide out

    DOI 10.1145/120694.120708. [BF94] G. Bjrck and R. Frberg. Methods to “divide out” certain solutions from systems of algebraic equations, applied to find all cyclic 8-roots. In Analysis, Algebra and Computers in Mathemat- ical Research, Proceedings of the twenty-first Nordic congress of mathematicians, edited by M. Gyllenberg and L. E. Persson, volume 564 of...

    work page doi:10.1145/120694.120708

  4. [4]

    Bliss, J

    [BSVY15] N. Bliss, J. Sommars, J. Verschelde, X. Yu. Solving polynomial systems in the cloud with polynomial homotopy continuation. In the Proceedings of the 17th International Workshop on Computer Algebra in Scientific Computing (CASC 2015), edited by V. P. Gerdt, W. Koepf, W. M. Seiler, and E. V. Vorozhtsov, volume 9301 of Lecture Notes in Computer Scien...

    work page 2015

  5. [5]

    DOI 10.1007/978-3-319-24021-3

    work page doi:10.1007/978-3-319-24021-3

  6. [6]

    [BT18] P

    DOI 10.1109/TVCG.2011.185. [BT18] P. Breiding and S. Timme. HomotopyContinuation.jl: A package for homotopy continuation in Julia. In the proceedings of ICMS 2018, the 6th International Conference on Mathematical Software, South Bend, IN, USA, July 24-27, 2018, edited by J. H. Davenport, M. Kauers, G. Labahn, and J. Urban, volume 10931 of Lecture Notes in...

    work page doi:10.1109/tvcg.2011.185 2011

  7. [7]

    [Chu06] W

    DOI 10.1007/978-3-319-96418-8. [Chu06] W. J. Chun. Core Python Programming. Prentice Hall, 2nd Edition,

    work page doi:10.1007/978-3-319-96418-8

  8. [8]

    [EM99] I.Z

    DOI 10.2140/gt.2018.22.1405. [EM99] I.Z. Emiris and B. Mourrain. Computer algebra methods for studying and computing molecular conformations. Algorithmica 25, pages 372402,

    work page doi:10.2140/gt.2018.22.1405 2018

  9. [9]

    [APP] explorable circle tangency https://github.com/JazzTap/mcs563/tree/master/ Apollonius] [HHM13] J

    DOI: 10.1007/PL00008283. [APP] explorable circle tangency https://github.com/JazzTap/mcs563/tree/master/ Apollonius] [HHM13] J. Hauenstein, Y.-H. He, and D. Mehta. Numerical elimination and moduli space of vacua. Journal of High Energy Physics,

    work page doi:10.1007/pl00008283

  10. [10]

    [HHS13] W

    DOI: 10.1007/JHEP09(2013)083. [HHS13] W. Hao, J. D. Hauenstein, C.-W. Shu, A. J. Sommese, Z. Xu, and Y.-T. Zhang. A homotopy method based on WENO schemes for solving steady state problems of hyper- bolic conservation laws. Journal of Computational Physics, 250, pages 332346

    work page doi:10.1007/jhep09(2013)083 2013

  11. [11]

    [HLB01] Y

    DOI: 10.1016/j.jcp.2013.05.008. [HLB01] Y. Hida, X. S. Li, and D. H. Bailey. Algorithms for quad-double precision floating point arith- metic. In the Proceedings of the 15th IEEE Symposium on Computer Arithmetic (Arith-15 2001), pages 155--162. IEEE Computer Society,

    work page doi:10.1016/j.jcp.2013.05.008 2013

  12. [12]

    [HCJL] A Julia package for solving systems of polynomials via homotopy continuation

    DOI 10.1109/ARITH.2001.930115. [HCJL] A Julia package for solving systems of polynomials via homotopy continuation. https:// github.com/JuliaHomotopyContinuation [Hun07] J. D. Hunter. Matplotlib: A 2D Graphics Environment. Computing in Science and Engineering 9(3): 90-95,

    work page doi:10.1109/arith.2001.930115 2001

  13. [13]

    D., 2007, @doi [Computing in Science and Engineering] 10.1109/MCSE.2007.55 , 9, 90

    DOI 10.1109/MCSE.2007.55. [GLW05] T. Gao, T.Y. Li, and M. Wu. Algorithm 846: MixedVol: a software package for mixed-volume computation. ACM Trans. Math. Softw., 31(4):555-560,

    work page doi:10.1109/mcse.2007.55 2007

  14. [14]

    [GBH16] E

    DOI 10.1145/1114268.1114274. [GBH16] E. Gross, D. Brent, K. L. Ho, D. J. Bates, and H. A. Harrington. Numerical algebraic geom- etry for model selection and its application to the life sciences. Journal of The Royal Society Interface, 13: 20160256

    work page doi:10.1145/1114268.1114274

  15. [15]

    [GHR16] E

    DOI: 10.1098/rsif.2016.0256. [GHR16] E. Gross, H. A. Harrington, Z. Rosen, and B. Sturmfels. Algebraic Systems Biology: A Case Study for the Wnt Pathway. Bulletin of Mathematical Biology 78, pages 2151,

    work page doi:10.1098/rsif.2016.0256 2016

  16. [16]

    [GPV13] E

    DOI: 10.1007/s11538-015-0125-1. [GPV13] E. Gross, S. Petrovi, and J. Verschelde. Interfacing with PHCpack. The Journal of Software for Algebra and Geometry: Macaulay2, 5:20-25,

    work page doi:10.1007/s11538-015-0125-1

  17. [17]

    [HS95] B

    DOI 10.2140/jsag.2013.5.20. [HS95] B. Huber and B. Sturmfels. A polyhedral method for solving sparse polynomial systems. Math- ematics of Computation, 64(212):1541-1555,

    work page doi:10.2140/jsag.2013.5.20 2013

  18. [18]

    [HSS98] B

    DOI 10.1090/S0025-5718-1995-1297471-4. [HSS98] B. Huber, F. Sottile, and B. Sturmfels. Numerical Schubert calculus. Journal of Symbolic Computation, 26(6):767-788,

    work page doi:10.1090/s0025-5718-1995-1297471-4 1995

  19. [19]

    13 [IPYW] ipywidgets: Interactive HTML Widgets https://github.com/jupyter-widgets/ ipywidgets [SymPy] D

    DOI 10.1006/jsco.1998.0239. 13 [IPYW] ipywidgets: Interactive HTML Widgets https://github.com/jupyter-widgets/ ipywidgets [SymPy] D. Joyner, O. ertk, A. Meurer, and B. E. Granger. Open source computer algebra sys- tems: SymPy. ACM Communications in Computer Algebra 45(4): 225-234 ,

    work page doi:10.1006/jsco.1998.0239 1998

  20. [20]

    [Pascal] JupyterHub deployment of phcpy

    DOI 10.1145/2110170.2110185. [Pascal] JupyterHub deployment of phcpy. Website, accessed May

    work page doi:10.1145/2110170.2110185

  21. [21]

    [KMC18] C

    DOI 10.3233/978-1-61499- 649-1-87. [KMC18] C. Knoll, D. Mehta, T. Chen, and F. Pernkopf. Fixed Points of Belief PropagationAn Analysis via Polynomial Homotopy Continuation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 40, pages 21242136,

    work page doi:10.3233/978-1-61499-

  22. [22]

    [Ley11] A

    DOI 10.1109/TPAMI.2017.2749575. [Ley11] A. Leykin. Numerical algebraic geometry. The Journal of Software for Algebra and Geometry: Macaulay2, 3:5-10,

    work page doi:10.1109/tpami.2017.2749575 2017

  23. [23]

    [LVZ06] A

    DOI 10.2140/jsag.2011.3.5. [LVZ06] A. Leykin, J. Verschelde, and A. Zhao. Newton’s method with deflation for isolated singu- larities of polynomial systems. Theoretical Computer Science, 359(1-3):111-122,

    work page doi:10.2140/jsag.2011.3.5 2011

  24. [24]

    [LLMM14] L

    DOI 10.1016/j.tcs.2006.02.018. [LLMM14] L. Liberti, C. Lavor, N. Maculan, and A. Mucherino. Euclidean Distance Geometry and Ap- plications. SIAM Review 56, no. 1 (January 2014):

    work page doi:10.1016/j.tcs.2006.02.018 2006

  25. [25]

    Liberti, B

    DOI 10.1137/120875909 [LML14] L. Liberti, B. Masson, J. Lee, C. Lavor, and A. Mucherino. On the number of realizations of certain henneberg graphs arising in protein conformation. Discrete Applied Mathematics, 165, page 213232,

    work page doi:10.1137/120875909

  26. [26]

    DOI: 10.1016/j.dam.2013.01.020. [M2] D. R. Grayson and M. E. Stillman. Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2 [MT08] T. Mizutani and A. Takeda. DEMiCs: A software package for computing the mixed volume via dynamic enumeration of all mixed cells. In Software for Algebraic Geometry, edited by M. E...

    work page doi:10.1016/j.dam.2013.01.020 2013

  27. [27]

    [MW90] A

    DOI 10.1007/978-0-387-78133-4. [MW90] A. P. Morgan and C. W. Wampler. Solving a Planar Four-Bar Design Using Continuation. Journal of Mechanical Design, 112(4): 544-550,

    work page doi:10.1007/978-0-387-78133-4

  28. [28]

    [NAG4M2] Branch NAG of M2 repository

    DOI 10.1115/1.2912644. [NAG4M2] Branch NAG of M2 repository. https://github.com/antonleykin/M2/tree/NAG [MSDB] MySQLdb 1.2.4b4 documentation https://mysqlclient.readthedocs.io/ [PHCPY] phcpy 0.9.5 documentation http://homepages.math.uic.edu/~jan/phcpy_doc_html/ 14 [Sage] The Sage Developers. SageMath, the Sage Mathematics Software System, Version 7.6 . ht...

    work page doi:10.1115/1.2912644

  29. [29]

    [SJ05] W

    DOI 10.5281/zenodo.820864. [SJ05] W. Stein and D. Joyner. Sage: System for algebra and geometry experimentation. ACM SIGSAM Bulletin 39(2): 61-64,

    work page doi:10.5281/zenodo.820864

  30. [30]

    [SWM16] H

    DOI 10.1145/1101884.1101889. [SWM16] H. Sidky, J. K. Whitmer, and D. Mehta. Reliable mixture critical point computation using polynomial homotopy continuation. AIChE Journal. Thermodynamics and Molecular-Scale Phenomena, 62(12): 4497-4507,

    work page doi:10.1145/1101884.1101889

  31. [31]

    [SVW03] A

    DOI 10.1002/aic.15319. [SVW03] A. J. Sommese, J. Verschelde, and C. W. Wampler. Numerical irreducible decomposition using PHCpack. In Algebra, Geometry and Software Systems, edited by M. Joswig and N. Takayama, pages 109-130, Springer-Verlag

    work page doi:10.1002/aic.15319

  32. [32]

    DOI 10.1007/978-3-662-05148-1

    work page doi:10.1007/978-3-662-05148-1

  33. [33]

    DOI 10.1007/3-540- 27357-3

    work page doi:10.1007/3-540-

  34. [34]

    [Ver14] J

    DOI 10.1145/317275.317286. [Ver14] J. Verschelde. Modernizing PHCpack through phcpy. Proceedings of the 6th European Confer- ence on Python in Science (EuroSciPy 2013), edited by P. de Buyl and N. Varoquaux, pages 71-76,

    work page doi:10.1145/317275.317286 2013

  35. [35]

    Verschelde

    [Ver18] J. Verschelde. A Blackbox Polynomial System Solver for Shared Memory Parallel Computers. In Computer Algebra in Scientific Computing, 20th International Workshop, CASC 2018, Lille, France, edited by V. P. Gerdt, W. Koepf, W. M. Seiler, and E. V. Vorozhtsov, volume 11077 of Lecture Notes in Computer Science, pages 361-375. Springer-Verlag,

    work page 2018

  36. [36]

    DOI 10.1007/978- 3-319-99639-4

    work page doi:10.1007/978-

  37. [37]

    [VVC94] J

    DOI 10.1007/BF01202036. [VVC94] J. Verschelde, P. Verlinden, and R. Cools. Homotopies exploiting Newton polytopes for solving sparse polynomial systems. SIAM Journal on Numerical Analysis 31(3):915-930,

    work page doi:10.1007/bf01202036

  38. [38]

    [VY15] J

    DOI 10.1137/0731049. [VY15] J. Verschelde and X. Yu. Polynomial Homotopy Continuation on GPUs. ACM Com- munications in Computer Algebra, volume 49, issue 4, pages 130-133,

    work page doi:10.1137/0731049

  39. [39]

    [WS11] C

    DOI 10.1145/2893803.2893810. [WS11] C. W. Wampler & A. J. Sommese Numerical algebraic geometry and algebraic kinematics. Acta Numerica, 20, pages 469567

    work page doi:10.1145/2893803.2893810

  40. [40]

    [Yu15] X

    DOI: 10.1017/S0962492911000067. [Yu15] X. Yu. Accelerating Polynomial Homotopy Continuation on Graphics Processing Units. PhD thesis, University of Illinois at Chicago,

    work page doi:10.1017/s0962492911000067