Pith. sign in

REVIEW 2 major objections 5 minor 110 references

Tube volumes around single-layer sigmoid decision boundaries grow only polynomially in network width, giving O(w^n/t) condition-number tails.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-10 08:34 UTC pith:3WIVMAK2

load-bearing objection Solid Pfaffian tube formula plus a genuine poly-in-width improvement for shallow rational sigmoids; the ball-containment and smoothness hyps are real but already flagged by the authors. the 2 major comments →

arxiv 2607.08370 v1 pith:3WIVMAK2 submitted 2026-07-09 math.AG cs.LG

Tubular Neighbourhoods of Pfaffian Sets and Applications to Neural Networks

Paul Lezeau , Martin Lotz This is my paper
classification math.AG cs.LG MSC 14P1068T0752A3914P15
keywords Pfaffian functionstubular neighbourhoodsGauss map degreeneural network classifierscondition numbersigmoid networksBernstein–Kushnirenko–Khovanskiifewnomials
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper bounds how much volume sits inside an ε-tube around a smooth hypersurface defined by Pfaffian functions—functions that obey triangular first-order PDEs with polynomial coefficients, including common neural activations such as sigmoid and tanh. Those volume bounds are stated purely in terms of the Pfaffian format of the defining function and recover classical algebraic tube estimates when the format is purely polynomial. Applied to neural classifiers, the same bounds become tail estimates for a condition number that measures relative distance to the decision boundary: large condition number means a small relative perturbation can flip the predicted class. For generic multi-layer Pfaffian nets the resulting constants still carry an exponential factor in the number of hidden units. The main improvement is for single-hidden-layer logistic networks whose first-layer weights are rational with bounded denominator: after an exponential substitution the Gauss-map system becomes a Laurent system whose Bernstein volume is polynomial in width, and the tube probability and condition-number tails therefore become polynomial rather than exponential in width.

Core claim

For a smooth compact hypersurface V = Z(f) given by a single-hidden-layer logistic network with rational first-layer weights of lattice constant L, every section degree of the Gauss map is at most K(n,L) w^{2n}. Consequently the uniform probability that a random point in a ball lies within distance ε of V is at most 2 K(n,L) w^{2n} [(1+ε/ρ)^n − 1], and the local condition-number tail decays as O(w^n / t).

What carries the argument

The maximal degree of the generalised Gauss map of a hypersurface (and of its generic affine sections). Bounding that degree by Khovanskii’s fewnomial count (general Pfaffian case) or by a Bernstein–Kushnirenko–Khovanskii volume after an exponential substitution (rational-weight single-layer case) converts classical integral-geometry tube formulae into explicit volume and condition-number estimates.

Load-bearing premise

Every pairwise decision boundary must sit entirely inside the ball on which the data are sampled, and the defining gradient must never vanish on that boundary; otherwise an exponential factor reappears or the surface is no longer smooth.

What would settle it

Construct a single-hidden-layer logistic network of width w with rational weights of fixed lattice constant L whose zero set is a smooth compact hypersurface whose Gauss-map degree grows faster than any constant times w^n, or whose ε-tube volume inside a containing ball exceeds the claimed polynomial bound for large w.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper derives volume bounds for tubular neighbourhoods of smooth bounded Pfaffian hypersurfaces in terms of the Pfaffian format (α, β, s), via Khovanskii bounds on the degrees of the generalised Gauss map of generic affine sections (Prop. 3.4, Thm. 3.6). These are applied to neural-network classifiers with Pfaffian activations to obtain uniform and Gaussian tail bounds on a local condition number C_p(X) = ∥X−p∥/dist(X, Σ) measuring distance to the decision boundary (Thms. 5.3–5.4, Cor. 5.7). For single-hidden-layer logistic networks with rational first-layer weights of lattice constant L, a Bernstein–Kushnirenko–Khovanskii count after an exponential chart yields mdeg(V) ≤ C(n,L) w^n (Prop. 4.9); a multiplicative-chart argument with a width-independent Pfaffian chain of length O(n) then bounds all section degrees by K(n,L) w^{2n} (Prop. 4.14), giving a polynomial-in-width tube formula (Thm. 4.15) and an O(w^n/t) condition-number tail (Cor. 5.8, Rem. 5.9). Multi-layer and singular cases are left as conjectures.

Significance. The work cleanly extends the algebraic tube-volume programme of Lotz and of Basu–Lerario to the Pfaffian setting that naturally contains sigmoid, tanh and related activations, and converts those bounds into explicit condition-number tails for neural classifiers. The single-layer rational-weight results are the strongest contribution: they replace the exponential Khovanskii factor 2^{w(w−1)/2} by a polynomial of degree 2n (with sharp leading order w^n for the top Gauss degree, Prop. 4.11), under explicitly stated hypotheses. All constants are expressed in terms of format or lattice data; the derivations are fully written and the limitations (ball containment, smoothness) are recorded honestly. This is a solid, technically careful contribution at the interface of real algebraic geometry and the geometric analysis of neural networks.

major comments (2)
  1. Thm. 4.15 and Cor. 5.8 require every pairwise decision boundary V_ij to lie inside the sampling ball B(p, ρ). Rem. 4.19 correctly notes that without this inclusion the sphere-boundary term ∂M = V ∩ S^{n−1}(p, ρ+ε) re-introduces the full Khovanskii factor 2^{w(w−1)/2} via the complete-intersection bound (3.3). The hypothesis is load-bearing for the polynomial-width claim that is the paper’s main selling point; the abstract and introduction should state it as prominently as the polynomial bound itself, and the Gaussian hybrid (Prop. 5.10) should be flagged as recovering only a hybrid (not fully polynomial) rate at fixed Gaussian scale (Rem. 5.11).
  2. Smoothness (∇f never vanishes on V) is indispensable for the Gauss-map degree to be well-defined and for the non-degeneracy lemma (Lem. 4.8) that justifies the BKK count. Decision boundaries of sigmoid networks can develop singularities for generic weights; the paper correctly excludes them by hypothesis and leaves the singular case open (§6.2). A short discussion of how restrictive this is in practice (or a pointer to the algebraic deformation strategy of Basu–Lerario and why it does not transfer) would strengthen the claims of applicability.
minor comments (5)
  1. Abstract and first paragraph of the introduction: the phrase “polynomial-in-width bounds for tubular neighbourhoods of the decision boundary” should be qualified by the ball-containment and smoothness hypotheses that make Thm. 4.15 possible.
  2. Notation: the same symbol σ is used for the logistic sigmoid and for the Gaussian standard deviation (e.g. Thm. 5.4, Prop. 5.10). A brief local reminder or a different letter for the variance would avoid momentary confusion.
  3. Prop. 4.9: the constant is stated as C(n,L) ≤ 2·n!(2L)^n, yet the BKK count already yields n!(2L)^n w^n; the extra factor 2 is harmless but could be tightened or explained.
  4. Example 5.12: the explicit multi-layer constant 6·2^{h(h−1)/2}(n(4ℓ+1)+2)^h is useful; a one-line comparison with the single-layer polynomial of Cor. 5.8 would help the reader appreciate the improvement.
  5. Typographical: “Khovanskii’s theorem [Kho91], Theorem 2.10” (p. 1) and a few similar double citations; also “the (1−δ)-quantile” in Rem. 5.2 could be written more carefully.

Circularity Check

0 steps flagged

No circularity: tube bounds and poly-in-w Gauss-map degrees are derived from Khovanskii/BKK and Weyl-type integral geometry, not from fitted parameters or self-definitional loops.

full rationale

The derivation chain is self-contained mathematics. Theorem 3.6 bounds tube volumes of smooth Pfaffian hypersurfaces by applying Khovanskii’s fewnomial bound (Thm 2.10) to the Gauss-map fibre system and feeding the resulting section degrees into a one-sided Weyl/Crofton estimate (Thm 3.2, generalising Lotz 2015). For single-hidden-layer logistic networks with rational first-layer weights, Prop. 4.9 replaces Khovanskii by a Bernstein–Kushnirenko–Khovanskii count on a Laurent system obtained after the exponential chart y_j = e^{-x_j/q}; the Newton polytope is a zonotope of lattice constant L, yielding mdeg(V) ≤ C(n,L) w^n. Prop. 4.14 recovers all lower section degrees by the same chart, now with a width-independent Pfaffian chain of length ≤ 2n (only logarithms of the section), so Khovanskii gives O(w^{2n}). Theorem 4.15 and Cor. 5.8 are direct substitutions of these degree bounds into the tube formula; the condition number C_p(X) = ||X-p||/Δ(X) is the classical relative distance to the decision boundary and is never used as an input to the volume estimates. No parameter is fitted to data, no uniqueness theorem is imported from the authors to forbid alternatives, and the sole self-citation (Lotz 2015) supplies the algebraic precursor that is being generalised, not a load-bearing unverified premise. The ball-containment and non-vanishing-gradient hypotheses are stated explicitly and limit the scope; they do not create a definitional loop. Score 0 is therefore the correct finding.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The paper rests on classical fewnomial and integral-geometric tools (Khovanskii, BKK, Weyl-type tube formulae) together with the modelling assumptions that activations are autonomous Pfaffian, first-layer weights are rational of bounded denominator, and decision boundaries are smooth and contained in a fixed ball. No numerical free parameters are fitted; the only ‘constants’ are explicit functions of format or lattice size. No new physical or geometric entities are postulated beyond the standard condition-number analogy.

axioms (6)
  • standard math Khovanskii’s fewnomial bound on the number of non-degenerate real solutions of a Pfaffian system (Thm. 2.10)
    Used throughout §§3–4 to control Gauss-map degrees and section degrees.
  • standard math Bernstein–Kushnirenko–Khovanskii mixed-volume bound for Laurent systems
    Central to the polynomial-width estimate after the exponential substitution (Prop. 4.9).
  • standard math Weyl-type integral-geometric tube formula relating volume of T(M,ε) to section degrees of the Gauss map (Thm. 3.2, citing Lotz 2015)
    Converts degree bounds into volume and probability statements.
  • domain assumption Activation functions are autonomous Pfaffian of fixed format (α,β,s) with s≥1
    Needed for the format calculus of Prop. 4.1 and for the chain length to be linear in the number of units.
  • domain assumption First-layer weight vectors are rational with common denominator q and lattice constant L = q max |a_ki|
    Essential for the integer lattice that makes the BKK count polynomial in width (Prop. 4.9).
  • domain assumption The hypersurface V (or each pairwise decision boundary) is smooth (∇f ≠ 0) and contained in a fixed ball B(p,ρ)
    Smoothness guarantees that the Gauss map is well-defined and degrees are finite; ball containment eliminates the sphere-boundary term that would re-introduce an exponential factor (Rem. 4.19).

pith-pipeline@v1.1.0-grok45 · 39023 in / 3296 out tokens · 35113 ms · 2026-07-10T08:34:32.122277+00:00 · methodology

0 comments
read the original abstract

We derive bounds for the volume of tubular neighbourhoods of smooth Pfaffian hypersurfaces, generalising known results for algebraic varieties. The bounds are given in terms of the Pfaffian format of the defining functions. As an application, we obtain tail bounds on the probability distribution of a condition number measuring the robustness of neural network classifiers with Pfaffian activation functions, in both the uniform and Gaussian settings. In the special case of single-hidden-layer sigmoid networks with rational weights, we derive polynomial-in-width bounds for tubular neighbourhoods of the decision boundary.

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

110 extracted references · 110 canonical work pages · 6 internal anchors

  1. [1]

    Algebraic Model Theory , editor=

    Marker, David , title=. Algebraic Model Theory , editor=. 1997 , publisher=. doi:10.1007/978-94-015-8923-9_8 , url=

    work page doi:10.1007/978-94-015-8923-9_8 1997

  2. [2]

    Parameterwise Sharpness of Khovanskii's Bezout-type Bound for Pfaffian Functions

    Terence Bickerton and Joseph Harrison and Olivia Hornakova and Dominic Le-Mar and Abhiram Natarajan and Nadia Potter , year=. On the Sharpness of. 2606.24373 , archivePrefix=

    work page internal anchor Pith review Pith/arXiv arXiv

  3. [3]

    2005 , publisher=

    Using algebraic geometry , author=. 2005 , publisher=

    work page 2005

  4. [4]

    Goodfellow and Rob Fergus , editor =

    Christian Szegedy and Wojciech Zaremba and Ilya Sutskever and Joan Bruna and Dumitru Erhan and Ian J. Goodfellow and Rob Fergus , editor =. Intriguing properties of neural networks , booktitle =. 2014 , url =

    work page 2014

  5. [5]

    Explaining and Harnessing Adversarial Examples

    Goodfellow, Ian J. and Shlens, Jonathon and Szegedy, Christian , title =. International Conference on Learning Representations , year =. 1412.6572 , archivePrefix =

    work page internal anchor Pith review Pith/arXiv arXiv

  6. [6]

    Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition , pages =

    Moosavi-Dezfooli, Seyed-Mohsen and Fawzi, Alhussein and Frossard, Pascal , title =. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition , pages =. 2016 , doi =

    work page 2016

  7. [7]

    Advances in Neural Information Processing Systems , volume =

    Fawzi, Alhussein and Moosavi-Dezfooli, Seyed-Mohsen and Frossard, Pascal , title =. Advances in Neural Information Processing Systems , volume =. 2016 , eprint =

    work page 2016

  8. [8]

    Advances in Neural Information Processing Systems , volume =

    Fawzi, Alhussein and Fawzi, Hamza and Fawzi, Omar , title =. Advances in Neural Information Processing Systems , volume =. 2018 , eprint =

    work page 2018

  9. [9]

    and Mahmoody, Mohammad , title =

    Mahloujifar, Saeed and Diochnos, Dimitrios I. and Mahmoody, Mohammad , title =. Proceedings of the AAAI Conference on Artificial Intelligence , volume =

    work page

  10. [10]

    2013 , publisher=

    Condition: The geometry of numerical algorithms , author=. 2013 , publisher=

    work page 2013

  11. [11]

    1993 , isbn =

    Macintyre, Angus and Sontag, Eduardo Daniel , title =. 1993 , isbn =. doi:10.1145/167088.167192 , booktitle =

    work page doi:10.1145/167088.167192 1993

  12. [12]

    Marek Karpinski and Angus Macintyre , title =. J. Comput. Syst. Sci. , volume =. 1997 , doi =

    work page 1997

  13. [13]

    IEEE transactions on neural networks and learning systems , volume=

    On the complexity of neural network classifiers: A comparison between shallow and deep architectures , author=. IEEE transactions on neural networks and learning systems , volume=. 2014 , publisher=

    work page 2014

  14. [14]

    and Thomas, Margaret E

    Jones, Gareth O. and Thomas, Margaret E. M. , journal=. The density of algebraic points on certain. 2012 , publisher=

    work page 2012

  15. [15]

    Gaussian Error Linear Units (

    Hendrycks, Dan and Gimpel, Kevin , journal=. Gaussian Error Linear Units (

    work page

  16. [16]

    Proceedings of the fourteenth international conference on artificial intelligence and statistics , pages=

    Deep sparse rectifier neural networks , author=. Proceedings of the fourteenth international conference on artificial intelligence and statistics , pages=. 2011 , organization=

    work page 2011

  17. [17]

    2007 , publisher=

    Random Fields and Geometry , author=. 2007 , publisher=

    work page 2007

  18. [18]

    Bifurcation of Planar Vector Fields and

    Roussarie, Robert , series=. Bifurcation of Planar Vector Fields and. 1998 , publisher=

    work page 1998

  19. [19]

    The Arnoldfest (Toronto, 1997) , series=

    On functions and curves defined by ordinary differential equations , author=. The Arnoldfest (Toronto, 1997) , series=. 1999 , publisher=

    work page 1997

  20. [20]

    Quantitative theory of ordinary differential equations and the tangential

    Yakovenko, Sergei , journal=. Quantitative theory of ordinary differential equations and the tangential

    work page

  21. [21]

    Foundations of Computational Mathematics , volume=

    Wilkinson's bus: Weak condition numbers, with an application to singular polynomial eigenproblems , author=. Foundations of Computational Mathematics , volume=. 2020 , publisher=

    work page 2020

  22. [22]

    Proceedings of the American Mathematical Society , volume=

    On the volume of tubular neighborhoods of real algebraic varieties , author=. Proceedings of the American Mathematical Society , volume=

    work page

  23. [23]

    Basu, Saugata and Lerario, Antonio , TITLE =. Math. Ann. , FJOURNAL =. 2023 , NUMBER =. doi:10.1007/s00208-022-02458-w , URL =

    work page doi:10.1007/s00208-022-02458-w 2023

  24. [24]

    Partitioning Theorems for Sets of Semi-

    Lotz, Martin and Natarajan, Abhiram and Vorobjov, Nicolai , journal=. Partitioning Theorems for Sets of Semi-

    work page

  25. [25]

    Quantitative study of semi-

    Zell, Thierry Paul , year=. Quantitative study of semi-

    work page

  26. [26]

    Khovanskii--

    Bates, Daniel J and Sottile, Frank , journal=. Khovanskii--. 2011 , publisher=

    work page 2011

  27. [27]

    2002 , PAGES =

    Marker, David , TITLE =. 2002 , PAGES =

    work page 2002

  28. [28]

    Computational Geometry , volume=

    Counting and cutting cycles of lines and rods in space , author=. Computational Geometry , volume=. 1992 , publisher=

    work page 1992

  29. [29]

    Prospects in mathematics (

    Wolff, Thomas , TITLE =. Prospects in mathematics (. 1999 , ISBN =

    work page 1999

  30. [30]

    Acta Arith

    Elekes, Gy\"orgy , TITLE =. Acta Arith. , FJOURNAL =. 1997 , NUMBER =

    work page 1997

  31. [31]

    Dvir, Zeev , TITLE =. Found. Trends Theor. Comput. Sci. , FJOURNAL =. 2010 , NUMBER =

    work page 2010

  32. [32]

    Electron

    Sharir, Micha and Sheffer, Adam and Solomon, Noam , TITLE =. Electron. J. Combin. , FJOURNAL =. 2016 , NUMBER =

    work page 2016

  33. [33]

    and Edelsbrunner, Herbert and Guibas, Leonidas J

    Clarkson, Kenneth L. and Edelsbrunner, Herbert and Guibas, Leonidas J. and Sharir, Micha and Welzl, Emo , TITLE =. Discrete Comput. Geom. , FJOURNAL =. 1990 , NUMBER =

    work page 1990

  34. [34]

    Pach, J\'anos and Sharir, Micha , TITLE =. Combin. Probab. Comput. , FJOURNAL =. 1998 , NUMBER =

    work page 1998

  35. [35]

    Proceedings of the

    Khovanskii, Askold G , TITLE =. Proceedings of the. 1984 , ISBN =

    work page 1984

  36. [36]

    , TITLE =

    Ilyashenko, Yu. , TITLE =. Bull. Amer. Math. Soc. (N.S.) , FJOURNAL =. 2002 , NUMBER =

    work page 2002

  37. [37]

    , TITLE =

    Szemer\'edi, Endre and Trotter, Jr., William T. , TITLE =. Combinatorica , FJOURNAL =. 1983 , NUMBER =

    work page 1983

  38. [38]

    SIAM Journal on Computing , volume =

    Aronov, Boris and Ezra, Esther and Zahl, Joshua , title =. SIAM Journal on Computing , volume =. 2020 , doi =

    work page 2020

  39. [39]

    SIAM Journal on Computing , volume=

    Efficient algorithm for generalized polynomial partitioning and its applications , author=. SIAM Journal on Computing , volume=. 2021 , publisher=

    work page 2021

  40. [40]

    , TITLE =

    Scanlon, T. , TITLE =. Gazette des Math\'ematiciens , NUMBER =. 2016 , PAGES =

    work page 2016

  41. [41]

    Duke Math

    Chernikov, Artem and Peterzil, Ya'acov and Starchenko, Sergei , TITLE =. Duke Math. J. , FJOURNAL =. 2024 , NUMBER =

    work page 2024

  42. [42]

    Lectures on discrete geometry , SERIES =

    Matou. Lectures on discrete geometry , SERIES =. 2002 , PAGES =

    work page 2002

  43. [43]

    Miller, Chris and Speissegger, Patrick , TITLE =. J. Symbolic Logic , FJOURNAL =. 2002 , NUMBER =

    work page 2002

  44. [44]

    Lion, Jean-Marie and Rolin, Jean-Philippe , TITLE =. Ann. Fac. Sci. Toulouse Math. (6) , FJOURNAL =. 1998 , NUMBER =

    work page 1998

  45. [45]

    The Journal of Symbolic Logic , author=

    Combinatorial Bounds in Distal Structures , DOI=. The Journal of Symbolic Logic , author=. 2023 , pages=

    work page 2023

  46. [46]

    1997 , publisher=

    Topology from the Differentiable Viewpoint , author=. 1997 , publisher=

    work page 1997

  47. [47]

    An Introduction to Manifolds , pages=

    Manifolds , author=. An Introduction to Manifolds , pages=. 2011 , publisher=

    work page 2011

  48. [48]

    2013 , publisher=

    Real algebraic geometry , author=. 2013 , publisher=

    work page 2013

  49. [49]

    Selecta Mathematica , volume=

    A theorem of the complement and some new o-minimal structures , author=. Selecta Mathematica , volume=. 1999 , publisher=

    work page 1999

  50. [50]

    Model completeness results for expansions of the ordered field of real numbers by restricted

    Wilkie, Alex , journal=. Model completeness results for expansions of the ordered field of real numbers by restricted

    work page

  51. [51]

    Speissegger, Patrick , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 1999 , PAGES =

    work page 1999

  52. [52]

    Geometric

    Grothendieck, Alexandre , TITLE =. Geometric. 1997 , ISBN =

    work page 1997

  53. [53]

    Ramsey growth in some

    Chernikov, Artem and Starchenko, Sergei and Thomas, Margaret EM , journal=. Ramsey growth in some. 2021 , publisher=

    work page 2021

  54. [54]

    Acta Mathematica , number =

    Larry Guth , title =. Acta Mathematica , number =. 2018 , doi =

    work page 2018

  55. [55]

    Journal of the American Mathematical Society , volume=

    A restriction estimate using polynomial partitioning , author=. Journal of the American Mathematical Society , volume=

    work page

  56. [56]

    Transactions of the American Mathematical Society , volume=

    Distinct distances in the complex plane , author=. Transactions of the American Mathematical Society , volume=

    work page

  57. [57]

    Geometric and Functional Analysis , volume=

    Joints of varieties , author=. Geometric and Functional Analysis , volume=. 2022 , publisher=

    work page 2022

  58. [58]

    Journal of the ACM , volume=

    Lower bounds for semialgebraic range searching and stabbing problems , author=. Journal of the ACM , volume=. 2023 , publisher=

    work page 2023

  59. [59]

    Simple proofs of classical theorems in discrete geometry via the

    Kaplan, Haim and Matou. Simple proofs of classical theorems in discrete geometry via the. Discrete & Computational Geometry , volume=. 2012 , publisher=

    work page 2012

  60. [60]

    On a real analog of

    Barone, Sal and Basu, Saugata , journal=. On a real analog of. 2016 , publisher=

    work page 2016

  61. [61]

    The Journal of Symbolic Logic , pages=

    Combinatorial bounds in distal structures , author=. The Journal of Symbolic Logic , pages=. 2023 , publisher=

    work page 2023

  62. [62]

    Incidences with

    Balsera, Alexander , journal=. Incidences with

    work page

  63. [63]

    Chernikov, Artem and Starchenko, Sergei , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2018 , NUMBER =

    work page 2018

  64. [64]

    Cutting lemma and

    Chernikov, Artem and Galvin, David and Starchenko, Sergei , journal=. Cutting lemma and. 2020 , publisher=

    work page 2020

  65. [65]

    Zarankiewicz’s problem for semilinear hypergraphs , volume=

    Basit, Abdul and Chernikov, Artem and Starchenko, Sergei and Tao, Terence and Tran, Chieu-Minh , year=. Zarankiewicz’s problem for semilinear hypergraphs , volume=. doi:10.1017/fms.2021.52 , journal=

    work page doi:10.1017/fms.2021.52 2021

  66. [66]

    An o-minimal

    Basu, Saugata and Raz, Orit E , journal=. An o-minimal. 2018 , publisher=

    work page 2018

  67. [67]

    2016 , publisher=

    Polynomial methods in combinatorics , author=. 2016 , publisher=

    work page 2016

  68. [68]

    Betti numbers of deterministic and random sets in semi-algebraic and o-minimal geometry

    Abhiram Natarajan. Betti numbers of deterministic and random sets in semi-algebraic and o-minimal geometry. 2020. doi:10.25394/PGS.12252185.v1

    work page doi:10.25394/pgs.12252185.v1 2020

  69. [69]

    Quantitative study of semi-Pfaffian sets

    Thierry Paul Zell. Quantitative study of semi-Pfaffian sets. 2003

    work page 2003

  70. [70]

    2022 , PAGES =

    Sheffer, Adam , TITLE =. 2022 , PAGES =

    work page 2022

  71. [71]

    Guth, Larry and Katz, Nets Hawk , journal=. On the. 2015 , publisher=

    work page 2015

  72. [72]

    Mathematische Zeitschrift , volume=

    The complexification and degree of a semi-algebraic set , author=. Mathematische Zeitschrift , volume=. 2002 , publisher=

    work page 2002

  73. [73]

    Discrete & Computational Geometry , volume=

    An incidence theorem in higher dimensions , author=. Discrete & Computational Geometry , volume=. 2012 , publisher=

    work page 2012

  74. [74]

    Generalizations of Joints Problem

    Generalizations of joints problem , author=. arXiv preprint arXiv:1606.08525 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  75. [75]

    Milnor, John , journal=. On the

    work page

  76. [76]

    Sur l’homologie des vari

    Thom, Ren. Sur l’homologie des vari. Differential and combinatorial topology , pages=

    work page

  77. [77]

    Izvestiya Rossiiskoi Akademii Nauk

    On the topology of real algebraic surfaces , author=. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya , volume=. 1949 , publisher=

    work page 1949

  78. [78]

    Effective cylindrical cell decompositions for restricted sub-

    Binyamini, Gal and Vorobjov, Nicolai , journal=. Effective cylindrical cell decompositions for restricted sub-. 2022 , publisher=

    work page 2022

  79. [79]

    Journal of Fixed Point Theory and Applications , volume=

    Polynomial partitioning for several sets of varieties , author=. Journal of Fixed Point Theory and Applications , volume=. 2017 , publisher=

    work page 2017

  80. [80]

    Mathematical Proceedings of the Cambridge Philosophical Society , volume=

    Polynomial partitioning for a set of varieties , author=. Mathematical Proceedings of the Cambridge Philosophical Society , volume=. 2015 , organization=

    work page 2015

Showing first 80 references.