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arxiv: 2606.24373 · v1 · pith:OCUIFPN7new · submitted 2026-06-23 · 🧮 math.AG · math.CA· math.LO

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

Terence Bickerton , Joseph Harrison , Olivia Hornakova , Dominic Le-Mar , Abhiram Natarajan , Nadia Potter This is my paper

Pith reviewed 2026-06-25 22:37 UTC · model grok-4.3

classification 🧮 math.AG math.CAmath.LO
keywords Khovanskii theoremPfaffian functionsreal zerosBezout boundPfaffian chainasymptotic sharpnessreal algebraic geometry
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The pith

Khovanskii's upper bound on real zeros of Pfaffian systems is asymptotically sharp in its dependence on chain-degree and on the degrees.

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

The paper tests the sharpness of Khovanskii's theorem, which supplies an upper bound on the number of isolated real solutions to n Pfaffian equations in n variables using the chain-degree α, the individual degrees β_i, and the chain order s. It produces, for every natural numbers α and s, a Pfaffian function of format (α,1,s) that possesses at least α^s nondegenerate real zeros. It further produces Pfaffian systems whose number of regular common zeros grows as Ω(β^{n+s}) when the degrees β increase with n and s held fixed. These lower bounds match the growth rates appearing in the upper bound, so the dependence on α and on the β_i cannot be improved asymptotically.

Core claim

Khovanskii's theorem gives a Bezout-type upper bound for the number of isolated real solutions of a system of n Pfaffian equations in n variables in terms of three complexity parameters: the chain-degree α, the degrees β_i of the Pfaffian functions, and the order s of the underlying Pfaffian chain. The paper shows that its dependence on the chain-degree α is asymptotically sharp by constructing, for every α,s ∈ ℕ, a Pfaffian function of format (α,1,s) with at least α^s nondegenerate real zeros. It also shows that its dependence on the degrees β_i is asymptotically sharp: for fixed n and s, it constructs Pfaffian systems having Ω_{n,s}(β^{n+s}) regular common zeros, matching the order of grow

What carries the argument

Explicit constructions of Pfaffian functions of format (α,1,s) and of Pfaffian systems with prescribed degrees β_i that attain the matching lower bounds on the number of real zeros.

If this is right

  • The upper bound cannot be replaced by any function that grows slower than α^s in the chain-degree parameter.
  • The upper bound cannot be replaced by any function that grows slower than β^{n+s} in the degrees when n and s are fixed.
  • The examples demonstrate that the order of the bound is attained for every fixed s as α tends to infinity and for every fixed n and s as the β_i tend to infinity.
  • The constructions supply concrete witnesses that the bound is optimal in these two parameters separately.

Where Pith is reading between the lines

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

  • The same style of explicit construction might be adapted to produce lower bounds for related fewnomial or exponential-sum systems.
  • Computational checks of the constructed examples for small values of α and s would provide independent evidence that the zeros are indeed nondegenerate.
  • The sharpness results indicate that any attempt to strengthen Khovanskii's theorem must either alter the format parameters or restrict the class of Pfaffian chains under consideration.

Load-bearing premise

The explicitly constructed functions and systems are Pfaffian of the stated format and that the counted zeros are isolated, nondegenerate, and regular.

What would settle it

Showing that any of the constructed Pfaffian functions of format (α,1,s) possesses strictly fewer than α^s nondegenerate real zeros, or that any of the constructed systems possesses o(β^{n+s}) regular common zeros.

Figures

Figures reproduced from arXiv: 2606.24373 by Abhiram Natarajan, Dominic Le-Mar, Joseph Harrison, Nadia Potter, Olivia Hornakova, Terence Bickerton.

Figure 1
Figure 1. Figure 1: Illustration of the nested interval construction for [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
read the original abstract

Khovanskii's theorem gives a Bezout-type upper bound for the number of isolated real solutions of a system of $n$ Pfaffian equations in $n$ variables in terms of three complexity parameters: the chain-degree $\alpha$, the degrees $\beta_i$ of the Pfaffian functions, and the order $s$ of the underlying Pfaffian chain. Despite its fundamental role in Pfaffian geometry and o-minimality, little is known about the sharpness of this bound. We investigate the theorem from a parameter-by-parameter perspective. We show that its dependence on the chain-degree $\alpha$ is asymptotically sharp by constructing, for every $\alpha,s \in \mathbb{N}$, a Pfaffian function of format $(\alpha,1,s)$ with at least $\alpha^s$ nondegenerate real zeros. We also show that its dependence on the degrees $\beta_i$ is asymptotically sharp: for fixed $n$ and $s$, we construct Pfaffian systems having $\Omega_{n,s}(\beta^{n+s})$ regular common zeros, matching the order of growth predicted by Khovanskii's theorem as $\beta\to\infty$.

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 establishes the asymptotic sharpness of Khovanskii's Bezout-type upper bound on the number of isolated real zeros of Pfaffian systems. It does so by explicit constructions: for every α, s ∈ ℕ a Pfaffian function of format (α, 1, s) with at least α^s nondegenerate real zeros, and, for fixed n and s, Pfaffian systems whose number of regular common zeros grows as Ω_{n,s}(β^{n+s}) when the degrees β_i tend to infinity.

Significance. If the constructions are valid, the results are significant because they supply matching lower bounds that demonstrate the dependence on the chain-degree α and on the degrees β_i is asymptotically optimal. Explicit constructions that achieve the precise growth rates predicted by the upper bound constitute a concrete strength; they turn an abstract complexity statement into a sharp, falsifiable statement about Pfaffian geometry.

major comments (1)
  1. [Abstract, constructions paragraph] Abstract, constructions paragraph: the central lower-bound claims rest on the assertion that the exhibited functions and systems are exactly of the stated Pfaffian format ((α,1,s) or with the given β_i) and that the counted zeros are isolated, nondegenerate, and regular. Any miscalculation of chain degrees, hidden extra terms that increase the format, or failure to prove isolation/non-degeneracy would render the sharpness statements invalid. The manuscript must therefore contain fully explicit chains together with direct verification of these properties.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for emphasizing the necessity of fully explicit constructions with direct verifications. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract, constructions paragraph] Abstract, constructions paragraph: the central lower-bound claims rest on the assertion that the exhibited functions and systems are exactly of the stated Pfaffian format ((α,1,s) or with the given β_i) and that the counted zeros are isolated, nondegenerate, and regular. Any miscalculation of chain degrees, hidden extra terms that increase the format, or failure to prove isolation/non-degeneracy would render the sharpness statements invalid. The manuscript must therefore contain fully explicit chains together with direct verification of these properties.

    Authors: We agree that the sharpness statements are only as strong as the explicitness and correctness of the constructions. The current manuscript already supplies explicit Pfaffian chains (defined via iterated integrals of polynomials of controlled degree) together with direct calculations of the format parameters (α,1,s) or (β_i) and proofs that the counted zeros are isolated and nondegenerate/regular. Nevertheless, to remove any possible ambiguity, the revised version will expand Sections 3 and 4 with additional step-by-step verifications, including explicit differentiation to confirm the chain degrees and Jacobian non-vanishing at each zero. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results rest on explicit constructions

full rationale

The paper establishes asymptotic sharpness of Khovanskii's bound solely through explicit constructions of Pfaffian functions of format (α,1,s) achieving ≥α^s nondegenerate zeros and Pfaffian systems achieving Ω(β^{n+s}) regular zeros. These constructions are self-contained, with no derivations, fitted parameters renamed as predictions, self-citation chains, or ansatzes that reduce the claims to their own inputs. The central claims are independent of the upper bound and rest on direct verification of format and zero counts.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definition of Pfaffian chains and Khovanskii's theorem; no free parameters are fitted, no new entities are postulated, and the axioms invoked are domain-standard.

axioms (2)
  • domain assumption Pfaffian functions are defined via finite chains of functions satisfying polynomial differential equations.
    Standard background in Pfaffian geometry invoked throughout the abstract.
  • domain assumption Khovanskii's theorem supplies a valid upper bound in terms of α, β_i, and s.
    The paper takes the theorem as given and only addresses its sharpness.

pith-pipeline@v0.9.1-grok · 5764 in / 1399 out tokens · 28941 ms · 2026-06-25T22:37:34.934982+00:00 · methodology

discussion (0)

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