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arxiv: 2110.03198 · v2 · submitted 2021-10-07 · 🧮 math.AG · math.CV· math.PR

The Expected Depth of Random Real Algebraic Plane Curves

Turgay Bayraktar , Al\.i Ula\c{s} \"Ozg\"ur K\.i\c{s}\.isel This is my paper

Pith reviewed 2026-05-24 12:43 UTC · model grok-4.3

classification 🧮 math.AG math.CVmath.PR
keywords random real algebraic curvesKostlan polynomialsexpected number of ovalsKac-Rice formulaasymptotic isotopyplane curvesreal algebraic geometry
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The pith

The expected number of ovals winding around any given point on a random real algebraic plane curve of even degree d equals √d/2.

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

The paper derives a Kac-Rice type formula that computes the expected number of two-sided components of a random real algebraic plane curve that wind around a fixed point. For even-degree Kostlan polynomials this expectation evaluates to √d/2 and stays the same regardless of where the point lies. A sympathetic reader cares because the count describes the typical nesting depth of the curve around points in the plane and supplies an exact asymptotic for the isotopy type in this random model.

Core claim

We obtain a Kac-Rice type formula that gives the expected number of two-sided components (i.e. ovals) of a random real algebraic plane curve winding around a given point. In particular, we show that expected number of such ovals for an even degree Kostlan polynomial is √d/2 and independent of the given point.

What carries the argument

Kac-Rice type formula applied to the random section of the line bundle that defines the Kostlan polynomials, counting only two-sided components winding around the fixed point.

Load-bearing premise

The Kac-Rice formula applies directly to the random section of the line bundle defining the Kostlan polynomials and correctly counts only the two-sided components that wind around the fixed point.

What would settle it

Generate many independent samples of even-degree Kostlan polynomials, count the ovals encircling a fixed interior point in each, and check whether the sample average converges to √d/2 as d grows.

read the original abstract

In this note we study asymptotic isotopy of random real algebraic plane curves. More precisely, we obtain a Kac-Rice type formula that gives the expected number of two-sided components (i.e.\ ovals) of a random real algebraic plane curve winding around a given point. In particular, we show that expected number of such ovals for an even degree Kostlan polynomial is $\frac{\sqrt{d}}{2}$ and independent of the given point.

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

1 major / 0 minor

Summary. The manuscript derives a Kac-Rice-type formula for the expected number of two-sided components (ovals) of a random real plane algebraic curve from the Kostlan ensemble that wind around a fixed point. For even degree d the expectation equals √d/2 and is independent of the point.

Significance. If the derivation is valid, the result supplies an exact, parameter-free closed-form expression for an isotopy invariant of random Kostlan curves. This is a precise quantitative statement in real algebraic geometry that could be checked numerically and extends known zero-counting results to a topologically filtered count.

major comments (1)
  1. [Abstract] Abstract, paragraph 2: the assertion that a standard Kac-Rice formula applied to the Kostlan line bundle directly yields the expected number of winding ovals requires an auxiliary indicator or integral that isolates components with nonzero winding number around the test point. The usual Kac-Rice density for real zeros of a Gaussian section does not perform this selection automatically; without an explicit derivation of that filtering step the claimed constant √d/2 cannot be verified and may fail when the point lies on a component or for odd-degree factors.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying a point that merits clarification. The major comment concerns the filtering mechanism in our Kac-Rice-type formula; we address it directly below and will revise the manuscript to make the auxiliary construction explicit.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 2: the assertion that a standard Kac-Rice formula applied to the Kostlan line bundle directly yields the expected number of winding ovals requires an auxiliary indicator or integral that isolates components with nonzero winding number around the test point. The usual Kac-Rice density for real zeros of a Gaussian section does not perform this selection automatically; without an explicit derivation of that filtering step the claimed constant √d/2 cannot be verified and may fail when the point lies on a component or for odd-degree factors.

    Authors: We agree that a plain Kac-Rice density on the zero set does not automatically select components by winding number, and that an auxiliary construction is required. Our derivation obtains the desired count by integrating the Kac-Rice density against a topological indicator (an integral representation of the winding number around the test point) that is well-defined for the Gaussian section of the Kostlan bundle. The resulting formula is stated and proved in Sections 2–3 of the manuscript; it evaluates to the constant √d/2 for even degree, independent of the base point. We will revise the abstract (and add a sentence in the introduction) to make this filtering step explicit and to cite the relevant sections. Events in which the test point lies on a component have measure zero and do not contribute to the expectation. The result is stated only for even degree, as required for the existence of closed ovals; the odd-degree case is outside the scope of the note. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external Kac-Rice formula to Kostlan ensemble

full rationale

The paper derives a Kac-Rice-type formula for the expected number of winding two-sided components around a fixed point and obtains the explicit constant √d/2 for even-degree Kostlan polynomials. The abstract and description present this as a direct application of the standard Kac-Rice formula to the Gaussian random section of the relevant line bundle, with no indication that the target count is defined in terms of itself, that parameters are fitted to the output quantity, or that the central claim reduces to a self-citation chain. The result is therefore self-contained against external mathematical machinery and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the Kac-Rice formula to the Kostlan Gaussian ensemble on the projective plane and on the definition of two-sided components; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Kac-Rice formula counts the expected number of zeros of a random section that satisfy the winding condition around a fixed point
    Invoked in the second sentence of the abstract as the tool that yields the formula.

pith-pipeline@v0.9.0 · 5612 in / 1194 out tokens · 30217 ms · 2026-05-24T12:43:31.107329+00:00 · methodology

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [1]

    Expected number and distribution of critica l points of real Lefschetz pencils

    Ancona, M. “Expected number and distribution of critica l points of real Lefschetz pencils.” Annales de l’Institut Fourier , Tome 70 (2020) no. 3, 1085-1113

    work page 2020

  2. [2]

    Topological propert ies of real algebraic varieties: du cot´ e de chez Rokhlin

    Degtyarev , A. I. , Kharlamov , V. M. “Topological propert ies of real algebraic varieties: du cot´ e de chez Rokhlin”, Russ. Math. Surv . 55 (2000), 735-814

    work page 2000

  3. [3]

    How many zeros of a random polyn omial are real?

    Edelman, A. , Kostlan, E. “How many zeros of a random polyn omial are real?”, Bulletin of the AMS vol 32 (1995), 1-37

    work page 1995

  4. [4]

    Pencils of cubics and algebraic curves in the real projective plane

    Fiedler-Le Touz´ e, S. “Pencils of cubics and algebraic curves in the real projective plane”, CRC Press (2009)

    work page 2009

  5. [5]

    On the number of connected components of random algebraic hypersur- faces

    Fyodorov , Y. , Lerario, A. , Lundberg, E. “On the number of connected components of random algebraic hypersur- faces ”, J. of Geometry and Physics, vol. 95 (2015), 1-20

    work page 2015

  6. [6]

    Expected topology of ran dom real algebraic submanifolds

    Gayet, D. , Welschinger , J.-Y. “Expected topology of ran dom real algebraic submanifolds”, J. Inst. Math. Jussieu 14(4) (2015), 673–702

    work page 2015

  7. [7]

    Lower estimates for the e xpected Betti numbers of random real hypersurfaces

    Gayet, D. , Welschinger , J.-Y. “Lower estimates for the e xpected Betti numbers of random real hypersurfaces”, J. London Math. Soc. (2) 90 (2014), 105–120

    work page 2014

  8. [8]

    Betti numbers of random r eal hypersurfaces and determinants of random symmetric matrices

    Gayet, D. , Welschinger , J.-Y. “Betti numbers of random r eal hypersurfaces and determinants of random symmetric matrices”, J. Eur . Math. Soc. 18 (2016), 733–772

    work page 2016

  9. [9]

    What is the total Betti nu mber of a random real hypersurface?

    Gayet, D. , Welschinger , J.-Y. “What is the total Betti nu mber of a random real hypersurface?”, J. reine angew . Math. 689 (2014), 137–168

    work page 2014

  10. [10]

    Exponential rarefacti on of real curves with many components

    Gayet, D. , Welschinger , J.-Y. “Exponential rarefacti on of real curves with many components”, Publ.math.IHES 113 , (2011) 69–96

    work page 2011

  11. [11]

    On the average number of real roots of a random al gebraic equation

    Kac, M. “On the average number of real roots of a random al gebraic equation.” Bull. Amer . Math. Soc. 49 (1943), 314-320

    work page 1943

  12. [12]

    Statistics on Hilbert’s 16t h problem

    Lerario, A. , Lundberg, E. “Statistics on Hilbert’s 16t h problem”, International Mathematics Research Notices, V ol. 2015, No. 12 (2014), 4293–4321

    work page 2015

  13. [13]

    Expected volume and Euler characteristi c of random submanifolds

    Letendre, T. “Expected volume and Euler characteristi c of random submanifolds”, J. Funct. Anal., 270 (2016), no. 8, 3047-3110

    work page 2016

  14. [14]

    On the number of nodal domains of random spherical harmonics

    Nazarov , F. , Sodin, M. “On the number of nodal domains of random spherical harmonics.”, Amer . J. Math. 131, (2009), 1337–1357

    work page 2009

  15. [15]

    Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions

    Nazarov , F. , Sodin, M. “ Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions”, Journal of Mathe matical Physics, Analysis, Geometry , vol. 12, No. 3 (2016), pp. 205–278

    work page 2016

  16. [16]

    Topologies of Nodal Sets of Rand om Band-Limited Functions

    Sarnak, P. , Wigman, I. “Topologies of Nodal Sets of Rand om Band-Limited Functions”, Comm. Pure and Appl. Math., Vol. LXXII (2019), 275-342

    work page 2019

  17. [17]

    Complexity of Bezout’s theorem II: Volumes and probabilities

    Shub, M. , Smale, S. “Complexity of Bezout’s theorem II: Volumes and probabilities”, Computational Algebraic Geometry (F. Eyssette and A. Galligo, eds.), Progr . Math., v ol. 109, (1993) Birkhauser , Boston, 267–285

    work page 1993

  18. [18]

    From the sixteenth Hilbert problem to tropica l geometry

    Viro, O. “From the sixteenth Hilbert problem to tropica l geometry” Japan J. Math. vol 3 (2008), 1-30

    work page 2008

  19. [19]

    Hilbert’s sixteenth problem

    Wilson, G. “Hilbert’s sixteenth problem”, Topology 17 (1978), 53-73. FACULTY OF ENGINEERING AND NATURAL SCIENCES , S ABANCI UNIVERSITY , ˙ISTANBUL , T URKEY Email address: tbayraktar@sabanciuniv.edu MATHEMATICS DEPARTMENT , M IDDLE EAST TECHNICAL UNIVERSITY Email address: akisisel@metu.edu.tr

    work page 1978