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arxiv: 2606.01415 · v1 · pith:CNN3XN3Snew · submitted 2026-05-31 · 🧮 math.AC

Implicitization of rational hypersurfaces by syzygies with respect to coefficient ideals

Falai Chen , Yunzhi Wang This is my paper

Pith reviewed 2026-06-28 15:45 UTC · model grok-4.3

classification 🧮 math.AC
keywords implicitizationsyzygiestoric varietiesrational hypersurfacescoefficient idealsCox ringbase pointsmatrix representations
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The pith

Matrix representations from syzygies restricted to a coefficient ideal implicitize rational hypersurfaces, with the determinant equaling a power of the implicit equation even for arbitrary base points in two dimensions.

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

The paper shows how to build matrices for implicitizing rational hypersurfaces that come from generically finite maps out of toric varieties. The matrices come from the coefficients of linear and quadratic syzygies of the defining equations, but only after restricting those coefficients to a chosen coefficient ideal J inside the Cox ring. In the special case of surfaces the resulting determinant is a power of the implicit equation no matter what the base points are, removing the usual requirement that the map be locally a complete intersection at those points. Readers interested in computational algebraic geometry would care because the method widens the range of parametrizations that can be implicitized by linear algebra alone.

Core claim

For a generically finite rational map from an n-dimensional toric variety to projective space of one higher dimension, the implicitization matrix assembled from linear and quadratic syzygies with coefficients taken from the coefficient ideal J has determinant equal to a power of the implicit equation of the image hypersurface; when the domain is two-dimensional this holds for arbitrary base points without any local complete intersection hypothesis.

What carries the argument

The coefficient ideal J inside the Cox ring of the toric variety, used to restrict the coefficients appearing in the syzygy modules before forming the implicitization matrices.

If this is right

  • The construction supplies explicit matrices for implicitization in every dimension.
  • In two dimensions the determinant identity holds without the local complete intersection condition at base points.
  • The technique extends several earlier implicitization results for surfaces.
  • It applies directly to maps whose base loci are arbitrary.

Where Pith is reading between the lines

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

  • The restriction to J might allow similar relaxations of local conditions when implicitizing in dimensions higher than two.
  • Once the syzygies are computed, the method could be implemented to handle parametrizations with more general base loci than previously possible.
  • The approach may combine with other toric-geometry tools that also work in the Cox ring.

Load-bearing premise

The generically finite map from the toric variety admits linear and quadratic syzygies that can be restricted to the coefficient ideal J while preserving the property that the determinant of the resulting matrix is a power of the implicit equation.

What would settle it

Take a concrete example of a generically finite rational map from a toric surface to projective three-space whose base point is not a local complete intersection, compute the restricted syzygy matrices, and verify whether their determinant equals a power of the actual implicit polynomial.

read the original abstract

We study rational hypersurfaces $\mathscr{S}$ defined as the closure of the image of a generically finite rational map $\phi:\mathscr{X}\rightarrow \mathbb{P}^{n+1}$, where $\mathscr{X}$ is an $n$-dimensional toric variety. We provide matrix representations for the implicitization of $\mathscr{S}$ that are constructed from the coefficients of linear syzygies and quadratic syzygies of the parametric equations. A central feature of our construction is the restriction of all coefficients in the Cox ring $R$ to a specific coefficient ideal $J$. In the two-dimensional case, this approach eliminates the need for $\phi$ to be locally a complete intersection at the base points, that is, the determinant of the implicitization matrix is equal to a power of the implicit equation for arbitrary base points. This result generalizes several previous results in surface implicitization.

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

0 major / 2 minor

Summary. The paper studies rational hypersurfaces defined as the closure of the image of a generically finite rational map ϕ from an n-dimensional toric variety X to P^{n+1}. It constructs matrix representations for the implicit equation of the hypersurface from the coefficients of linear and quadratic syzygies of the parametric equations, with all coefficients restricted to a coefficient ideal J in the Cox ring R. The central result is that, in the two-dimensional case, the determinant of the resulting implicitization matrix equals a power of the implicit equation even when ϕ is not locally a complete intersection at the base points, thereby generalizing several prior surface implicitization results.

Significance. If the construction and determinant claim hold, the work provides a direct generalization of existing syzygy-based implicitization methods that removes the locally complete intersection hypothesis at base points for surfaces. This broadens applicability to maps with arbitrary base points and could streamline computations in algebraic geometry and computer algebra systems. The explicit restriction to the coefficient ideal J is a clean technical device that merits attention if verified.

minor comments (2)
  1. [Abstract] The abstract and introduction should include a brief statement on the existence assumptions for the linear and quadratic syzygies whose coefficients are restricted to J; while taken as given for the maps under study, an explicit reference to the relevant proposition or hypothesis would aid readability.
  2. Consider adding a short computational example (e.g., a concrete toric surface map with non-LCI base points) to illustrate that the determinant indeed yields a power of the implicit equation; this would strengthen the presentation without altering the central claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report lists no specific major comments, so there are no individual points requiring point-by-point rebuttal or revision.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained construction

full rationale

The paper constructs implicitization matrices directly from the coefficients of linear and quadratic syzygies of the parametric equations, with coefficients restricted to the coefficient ideal J in the Cox ring. The 2D claim that the determinant equals a power of the implicit equation for arbitrary base points follows from this explicit matrix construction under the generically finite map assumption, without any reduction to fitted parameters, self-definitions, or load-bearing self-citations that themselves depend on the target result. The generalization of prior surface results is presented as an extension of the same syzygy-based method rather than a renaming or imported uniqueness theorem. No quoted step equates a prediction to its input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on standard toric-variety and syzygy assumptions in algebraic geometry; no free parameters or invented entities are visible from the abstract.

axioms (2)
  • domain assumption ϕ is a generically finite rational map from an n-dimensional toric variety X to P^{n+1}
    Stated as the setup for the hypersurface S.
  • domain assumption Linear and quadratic syzygies exist whose coefficients lie in the coefficient ideal J
    Central to the matrix construction described.

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Works this paper leans on

41 extracted references · 35 canonical work pages

  1. [1]

    and Huneke, Craig , TITLE =

    Aberbach, Ian M. and Huneke, Craig , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1996 , NUMBER =. doi:10.1090/S0002-9939-96-03058-4 , URL =

    work page doi:10.1090/s0002-9939-96-03058-4 1996

  2. [2]

    and Hoffman, J

    Adkins, William A. and Hoffman, J. William and Wang, Hao Hao , TITLE =. J. Symbolic Comput. , FJOURNAL =. 2005 , NUMBER =. doi:10.1016/j.jsc.2004.09.007 , URL =

    work page doi:10.1016/j.jsc.2004.09.007 2005

  3. [3]

    Botbol, Nicol\'as and Bus\'e, Laurent and Chardin, Marc , TITLE =. J. Algebra , FJOURNAL =. 2014 , PAGES =. doi:10.1016/j.jalgebra.2014.07.028 , URL =

    work page doi:10.1016/j.jalgebra.2014.07.028 2014

  4. [4]

    Bus\'e, Laurent and Chardin, Marc , TITLE =. J. Symbolic Comput. , FJOURNAL =. 2005 , NUMBER =. doi:10.1016/j.jsc.2004.04.005 , URL =

    work page doi:10.1016/j.jsc.2004.04.005 2005

  5. [5]

    Botbol, Nicol\'as and Chardin, Marc , TITLE =. J. Algebra , FJOURNAL =. 2017 , PAGES =. doi:10.1016/j.jalgebra.2016.11.017 , URL =

    work page doi:10.1016/j.jalgebra.2016.11.017 2017

  6. [6]

    Bus\'e, Laurent and Chen, Falai , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2021 , NUMBER =. doi:10.1090/tran/8358 , URL =

    work page doi:10.1090/tran/8358 2021

  7. [7]

    Bus\'e, Laurent and Cox, David and D'Andrea, Carlos , TITLE =. J. Algebra Appl. , FJOURNAL =. 2003 , NUMBER =. doi:10.1142/S0219498803000489 , URL =

    work page doi:10.1142/s0219498803000489 2003

  8. [8]

    Bus\'e, Laurent and Chardin, Marc and Jouanolou, Jean-Pierre , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2009 , NUMBER =. doi:10.1090/S0002-9939-09-09550-1 , URL =

    work page doi:10.1090/s0002-9939-09-09550-1 2009

  9. [9]

    Bus\'e, Laurent and Chardin, Marc and Simis, Aron , TITLE =. J. Algebra , FJOURNAL =. 2010 , NUMBER =. doi:10.1016/j.jalgebra.2010.07.006 , URL =

    work page doi:10.1016/j.jalgebra.2010.07.006 2010

  10. [10]

    Botbol, Nicol\'as and Dickenstein, Alicia , TITLE =. J. Symbolic Comput. , FJOURNAL =. 2016 , PAGES =. doi:10.1016/j.jsc.2015.09.001 , URL =

    work page doi:10.1016/j.jsc.2015.09.001 2016

  11. [11]

    Commutative algebra , PAGES =

    Botbol, Nicol\'as and Dickenstein, Alicia and Schenck, Hal , TITLE =. Commutative algebra , PAGES =. [2021] 2021 , ISBN =. doi:10.1007/978-3-030-89694-2\_3 , URL =

    work page doi:10.1007/978-3-030-89694-2 2021

  12. [12]

    and Eisenbud, David , TITLE =

    Buchsbaum, David A. and Eisenbud, David , TITLE =. Advances in Math. , FJOURNAL =. 1975 , NUMBER =. doi:10.1016/0001-8708(75)90046-8 , URL =

    work page doi:10.1016/0001-8708(75)90046-8 1975

  13. [13]

    1993 , PAGES =

    Bruns, Winfried and Herzog, J\"urgen , TITLE =. 1993 , PAGES =

    1993

  14. [14]

    Bus\'e, Laurent and Jouanolou, Jean-Pierre , TITLE =. J. Algebra , FJOURNAL =. 2003 , NUMBER =. doi:10.1016/S0021-8693(03)00181-9 , URL =

    work page doi:10.1016/s0021-8693(03)00181-9 2003

  15. [15]

    Botbol, Nicol\'as , TITLE =. J. Algebra , FJOURNAL =. 2011 , PAGES =. doi:10.1016/j.jalgebra.2011.09.019 , URL =

    work page doi:10.1016/j.jalgebra.2011.09.019 2011

  16. [16]

    Comput.-Aided Des

    Bus\'e, Laurent , TITLE =. Comput.-Aided Des. , FJOURNAL =. 2014 , PAGES =. doi:10.1016/j.cad.2013.08.014 , URL =

    work page doi:10.1016/j.cad.2013.08.014 2014

  17. [17]

    Cox, David and Goldman, Ronald and Zhang, Ming , TITLE =. J. Symbolic Comput. , FJOURNAL =. 2000 , NUMBER =. doi:10.1006/jsco.1999.0325 , URL =

    work page doi:10.1006/jsco.1999.0325 2000

  18. [18]

    Algebra Number Theory , FJOURNAL =

    Chardin, Marc , TITLE =. Algebra Number Theory , FJOURNAL =. 2013 , NUMBER =. doi:10.2140/ant.2013.7.1 , URL =

    work page doi:10.2140/ant.2013.7.1 2013

  19. [19]

    and Little, John B

    Cox, David A. and Little, John B. and Schenck, Henry K. , TITLE =. 2011 , PAGES =. doi:10.1090/gsm/124 , URL =

    work page doi:10.1090/gsm/124 2011

  20. [20]

    , TITLE =

    Cox, David A. , TITLE =. J. Algebraic Geom. , FJOURNAL =. 1995 , NUMBER =

    1995

  21. [21]

    Equations of parametric curves and surfaces via syzygies , BOOKTITLE =

    Cox, David A. Equations of parametric curves and surfaces via syzygies , BOOKTITLE =. 2001 , ISBN =. doi:10.1090/conm/286/04751 , URL =

    work page doi:10.1090/conm/286/04751 2001

  22. [22]

    Cox, David and Schenck, Hal , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2003 , NUMBER =. doi:10.1090/S0002-9939-02-06804-1 , URL =

    work page doi:10.1090/s0002-9939-02-06804-1 2003

  23. [23]

    , TITLE =

    Chen, Falai and Zheng, Jianmin and Sederberg, Thomas W. , TITLE =. Comput. Aided Geom. Design , FJOURNAL =. 2001 , NUMBER =. doi:10.1016/S0167-8396(01)00012-7 , URL =

    work page doi:10.1016/s0167-8396(01)00012-7 2001

  24. [24]

    [FKsd] Sergey Fomin and Anatol N

    Eisenbud, David , TITLE =. 1995 , PAGES =. doi:10.1007/978-1-4612-5350-1 , URL =

    work page doi:10.1007/978-1-4612-5350-1 1995

  25. [25]

    1998 , PAGES =

    Fulton, William , TITLE =. 1998 , PAGES =. doi:10.1007/978-1-4612-1700-8 , URL =

    work page doi:10.1007/978-1-4612-1700-8 1998

  26. [26]

    Gelfand, I. M. and Kapranov, M. M. and Zelevinsky, A. V. , TITLE =. 2008 , PAGES =

    2008

  27. [27]

    and Stillman, Michael E

    Grayson, Daniel R. and Stillman, Michael E. , TITLE =

  28. [28]

    L_2 - C ohomology of pseudoconvex domains with complete K \"ahler metric

    Huneke, Craig and Swanson, Irena , TITLE =. Michigan Math. J. , FJOURNAL =. 1995 , NUMBER =. doi:10.1307/mmj/1029005163 , URL =

    work page doi:10.1307/mmj/1029005163 1995

  29. [29]

    Integral closure of ideals, rings, and modules , SERIES =

    Huneke, Craig and Swanson, Irena. Integral closure of ideals, rings, and modules , SERIES =. 2006 , PAGES =

    2006

  30. [30]

    and Simis, A

    Herzog, J. and Simis, A. and Vasconcelos, W. V. , TITLE =. Commutative algebra (. 1983 , ISBN =

    1983

  31. [31]

    William and Wang, Hao Hao , TITLE =

    Hoffman, J. William and Wang, Hao Hao , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2006 , NUMBER =. doi:10.1090/S0002-9947-06-04119-5 , URL =

    work page doi:10.1090/s0002-9947-06-04119-5 2006

  32. [32]

    William and Wang, Haohao , TITLE =

    Hoffman, J. William and Wang, Haohao , TITLE =. J. Algebra Appl. , FJOURNAL =. 2023 , NUMBER =. doi:10.1142/S0219498823500688 , URL =

    work page doi:10.1142/s0219498823500688 2023

  33. [33]

    2022 , issue_date =

    Jia, Xiaohong and Chen, Falai and Yao, Shanshan , title =. 2022 , issue_date =. doi:10.1145/3551387 , journal =

    work page doi:10.1145/3551387 2022

  34. [34]

    Lai, Yisheng and Chen, Falai and Shi, Xiaoran , TITLE =. Comput. Aided Geom. Design , FJOURNAL =. 2019 , PAGES =. doi:10.1016/j.cagd.2019.03.001 , URL =

    work page doi:10.1016/j.cagd.2019.03.001 2019

  35. [35]

    Lai, Yisheng and Chen, Falai and Shi, Xiaoran and Gao, Yu , TITLE =. Comput. Aided Geom. Design , FJOURNAL =. 2022 , PAGES =. doi:10.1016/j.cagd.2021.102051 , URL =

    work page doi:10.1016/j.cagd.2021.102051 2022

  36. [36]

    Lipman, Joseph , TITLE =. Math. Res. Lett. , FJOURNAL =. 1994 , NUMBER =. doi:10.4310/MRL.1994.v1.n6.a10 , URL =

    work page doi:10.4310/mrl.1994.v1.n6.a10 1994

  37. [37]

    and Chen, Falai , TITLE =

    Sederberg, Thomas W. and Chen, Falai , TITLE =. Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH '95) , VOLUME =. 1995 , PAGES =. doi:10.1145/218380.218460 , URL =

    work page doi:10.1145/218380.218460 1995

  38. [38]

    Schenck, Hal and Seceleanu, Alexandra and Validashti, Javid , TITLE =. Math. Comp. , FJOURNAL =. 2014 , NUMBER =. doi:10.1090/S0025-5718-2013-02764-0 , URL =

    work page doi:10.1090/s0025-5718-2013-02764-0 2014

  39. [39]

    , TITLE =

    Vasconcelos, Wolmer V. , TITLE =. 1994 , PAGES =. doi:10.1017/CBO9780511574726 , URL =

    work page doi:10.1017/cbo9780511574726 1994

  40. [40]

    Linear Algebra Appl

    Weaver, Matthew , TITLE =. Linear Algebra Appl. , FJOURNAL =. 2025 , PAGES =. doi:10.1016/j.laa.2025.04.020 , URL =

    work page doi:10.1016/j.laa.2025.04.020 2025

  41. [41]

    and Chionh, Eng-Wee and Cox, David A

    Zheng, Jianmin and Sederberg, Thomas W. and Chionh, Eng-Wee and Cox, David A. , TITLE =. Topics in algebraic geometry and geometric modeling , SERIES =. 2003 , ISBN =. doi:10.1090/conm/334/05980 , URL =

    work page doi:10.1090/conm/334/05980 2003