REVIEW 3 major objections 4 minor 18 references
Smooth rigidity and Remez-type inequalities
T0 review · 3 major / 4 minor · reviewed 2026-08-27 · deepseek-v4-flash
Pith's one-line read A finite zero set's spacing controls the size of high derivatives of any smooth function vanishing on it.
desk verdict The finite-set upper bound is a promising new idea, but the proof has a real gap: the bump construction never verifies that the corrected function has C0 norm 1. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing objects are the rigidity constant $RG_d(Z)$ and the inverse Remez constant $\hat{R}_d(Z)$. Here $RG_d(Z)$ is the infimum of $M_{d+1}(f)$ over $C^{d+1}$ functions on the unit ball that vanish on $Z$ and satisfy $M_0(f)=1$, while $\hat{R}_d(Z)=1/R_d(Z)$, where $R_d(Z)$ is the smallest constant $K$ for which every real degree-$d$ polynomial's maximum on the ball is at most $K$ times its maximum on $Z$. The argument's mechanism is a smooth version of the Remez inequality, quoted from [18], which bounds $M_0(f)$ by a combination of the values of $f$ on $Z$ and the Taylor remainder, together with a bump-correction construction in which an extremal polynomial is modified locally around each point of $Z$. The invariant $\omega_d(Z)$, computed from covering numbers of $Z$ by cubes of size $\varepsilon$ minus a polynomial in $1/\varepsilon$, converts these estimates into explicit lower bounds.
What would settle it
Take the planar triangle $Z_h = \{(-1/2,0),(0,h),(1/2,0)\}$ and compute, for small $h$, the actual rigidity constant $RG_1(Z_h)$ by minimizing $M_2(f)$ over $C^2$ functions vanishing on $Z_h$ with $M_0(f)=1$. The theorem predicts $h/2 \le RG_1(Z_h) \le C/h$; if a direct computation found $RG_1(Z_h)$ decaying faster than $h$ or staying above $C/h$, the claimed equivalence would be false. A simpler check: for the constructed $f(x)=P(x)-\sum_j P(z_j)\psi(2(x-z_j)/\rho)$, compute $M_0(f)$; if it exceeds 1 for any finite $Z$, the upper-bound construction is not admissible.
Extended reading notes
Core claim
The paper's central claim is a two-sided comparison. For every subset $Z$ of the unit ball, $\frac{(d+1)!}{2}\hat{R}_d(Z) \le RG_d(Z)$; and for every finite $Z$ whose points are mutually at distance at least $\rho$, $RG_d(Z) \le C(n,d)\rho^{-(d+1)}\hat{R}_d(Z)$. The lower bound is obtained from a smooth Remez inequality quoted from [18], and the upper bound by taking a degree-$d$ polynomial $P$ that is extremal for $Z$ and subtracting small bump functions centered at the points of $Z$ to force $f$ to vanish there. A companion result characterizes the degenerate case: $RG_d(Z)=0$ exactly when $Z$ lies in the zero set of a degree-$d$ polynomial, in which case $\hat{R}_d(Z)=0$ as well. Additional corollaries give explicit lower bounds for finite and measurable sets through the geometric invariant $\omega_d(Z)$, including point configurations where no straight line meets more than two points.
Load-bearing premise
The two-sided estimate depends on the quoted smooth Remez inequality of [18] for the lower bound and on the unverified assumption that subtracting the bump corrections from the extremal polynomial produces a function whose maximum modulus is still 1; if either fails, the comparison does not follow.
Editorial extensions
If this is right
- The one-dimensional rigidity bound is a special case: for a set of $d+1$ separated points, $\hat{R}_d(Z)$ is positive, and Theorem 4.1 recovers a lower bound of order $\frac{(d+1)!}{2}$ times that inverse Remez constant.
- For finite sets with separation bounded below, the rigidity constant and the inverse Remez constant vanish and are positive together, and differ by at most a factor depending only on $n$, $d$, and $\rho$.
- In dimensions $n\ge 2$ there exist finite zero sets with arbitrarily small positive rigidity, so the minimal nonzero rigidity is no longer a universal constant as it is in one variable.
- For measurable sets with volume fraction $\lambda$, rigidity is at least $\frac{(d+1)!}{2}(\lambda/(4n))^d$, an explicit lower bound coming from volume alone.
- For finite sets whose cardinality $M$ exceeds $(4d)^n\rho^{-(n-1)}$, rigidity is bounded below by $\frac{(d+1)!}{2}\big((M\rho^n-(4d)^n\rho)/(4n)\big)^d$, covering dense grids and near-grids that no line crosses in more than two points.
Reading between the lines
- One consequence the author leaves implicit is that the rigidity constant can be estimated by finite-dimensional computation: for a fixed finite $Z$, $\hat{R}_d(Z)$ is a norming constant of a finite-dimensional polynomial space, so the two-sided bound turns an analytic infimum into a tractable linear-algebra problem.
- The open gap question suggests a testable classification: sets with empty interior but positive Hausdorff dimension, such as Cantor-type sets, may behave like finite sets if they are sufficiently spread out, or like interior sets if they contain accumulation directions; the paper's methods do not settle this.
- The bump-correction upper bound is likely not sharp; choosing $P$ and the bump profile so that the corrected function still has maximum exactly 1 would improve the constant $C(n,d)$ and may extend the equivalence to sets with less separation.
- For near-grid configurations, the theorem implies that density alone, not collinearity, controls rigidity; this could be relevant to numerical differentiation and interpolation on scattered point sets.
Formalized claims in Lean
-
Claim #1: The paper's central claim is a two-sided comparison. For every subset $Z$ of the unit ball, $\frac{(d+1)!}{2}\hat{R}_d(Z) \le RG_d(Z)$; and for every finite $Z$ whose points are mutually at distance at least $\rho$, $RG_d(Z) \le C(n,d)\rho^{-(d+1)}\hat{R}_d(Z)$. The lower bound is obtained from a smooth Remez inequality quoted from [18], and the upper bound by taking a degree-$d$ polynomial $P$ th
/-- @claim 1 The paper's central claim is a two-sided comparison. For every subset $Z$ of the unit ball, $\frac{(d+1)!}{2}\hat{R}_d(Z) \le RG_d(Z)$; and for every finite $Z$ whose points are mutually at distance at least $\rho$, $RG_d(Z) \le C(n,d)\rho^{-(d+1)}\hat{R}_d(Z)$. The lower bound is obtained from a smooth Remez inequality quoted from [18], and the upper bound by taking a degree-$d$ polynomial $P$ th -/ def central_claim : Prop :=
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper defines a d-rigidity constant RG_d(Z) for a subset Z of the unit ball B^n: the infimum of M_{d+1}(f), the norm of the (d+1)-st derivative, over C^{d+1} functions that vanish on Z and satisfy M_0(f)=1. It also recalls the (inverse) Remez constant \hat R_d(Z) and relates the two notions. The main results are Theorem 4.1 (lower bound RG_d(Z) \ge (d+1)!/2 \cdot \hat R_d(Z)), Theorem 4.2 (for finite sets with minimal separation \rho, the upper bound RG_d(Z) \le C(n,d) \hat R_d(Z)/\rho^{d+1}), and Theorem 4.3 (a lower bound in terms of cardinality and separation via the quantity \omega_d(Z)). The paper also gives examples, including near-grids and plane triangles, and discusses open questions about the gap between rigidity and inverse Remez constants.
Significance. If the upper-bound proof were completed, the paper would establish a useful bridge between finite-zero-set geometry and lower bounds on high-order derivatives in several variables, a setting where the one-dimensional interpolation tools do not apply. The lower-bound half, however, is not new: Theorem 4.1 is a direct consequence of [18, Theorem 5.2] combined with R_d(Z) \ge 1. The genuinely new contribution is the finite-set upper bound and the metric-geometric examples. The paper is clearly written and the definitions are natural, but the main proof of the upper bound has a genuine gap, and Proposition 5.3 contains two concrete errors. These issues affect the central equivalence claim and the stated form of Theorem 4.3.
major comments (3)
- [Section 5, proof of Theorem 4.2] The function f constructed by subtracting bumps from P is never shown to be admissible for the rigidity constant. RG_d(Z) is defined as the infimum over functions vanishing on Z with M_0(f)=1, but the proof bounds only M_{d+1}(f). If M_0(f)<1, the normalized function f/M_0(f) has M_{d+1} equal to M_{d+1}(f)/M_0(f), and no lower bound for M_0(f) is given. Cancellation between P and the bumps inside the support balls can reduce |f| below 1; for instance, a point where |P|=1 may lie inside one of the balls B_j, so the subtraction there is not forced to vanish. Consequently the right-hand inequality of Theorem 4.2 is not established. Since this is the load-bearing step for the claimed approximate equivalence for finite sets, the proof needs an additional argument (or an alternative construction that guarantees M_0(f)=1, or at least a lower bound for M_0(f) that is uniform in \rho and \hat R_d(Z)).
- [Section 5, Proposition 5.3] The substitution \epsilon = \rho in the definition of \omega_d(Z) is not valid for the stated counting. Since M(\epsilon,Z) is the minimum number of closed \epsilon-balls (in the l^\infty metric) needed to cover Z, and \rho is the minimal distance between points of Z, a ball of radius \rho can contain many \rho-separated points; even two points at distance \rho can be covered by one ball of radius \rho/2, let alone \rho. Thus M(\rho,Z) need not equal M=|Z|, and the inequality \omega_d(Z) \ge \rho^n (M - M_{n,d}(\rho)) has no basis. To obtain a covering count equal to M one would need \epsilon < \rho/2, which changes the resulting bound and the hypotheses of Theorem 4.3.
- [Section 5, Proposition 5.3 and Theorem 4.3] The proof also drops the normalization by the volume m_n(B^n). Theorem 3.2 gives \hat R_d(Z) \ge (\bar\lambda/(4n))^d with \bar\lambda = \omega_d(Z)/m_n(B^n). Hence after lower-bounding \omega_d(Z), the denominator should be m_n(B^n) \cdot 4n, not 4n. Since m_n(B^n) is not equal to 1 in general, Theorem 4.3 as stated is quantitatively incorrect. This is a separate error from the covering-number issue above, and both must be fixed before Theorem 4.3 can be accepted.
minor comments (4)
- [Section 5, proof of Theorem 4.2] The case \hat R_d(Z)=0, i.e., Z is contained in a degree-d polynomial zero set, is not treated in the proof. In that case one cannot choose a polynomial P with M_0(P)=1 and max_Z |P|=0; the upper bound is then trivial by Lemma 4.1 but this should be stated explicitly.
- [Introduction and Section 4] The abstract and introduction describe both inequalities as main results of the paper, but Theorem 4.1 is an immediate corollary of [18, Theorem 5.2] together with R_d(Z) \ge 1. The new contribution should be described more precisely as the finite-set upper bound and the geometric examples, with the dependence on [18] made explicit in the introduction.
- [Section 3 and Section 4] The paper switches between the Euclidean unit ball B^n and the l^\infty metric used in Definition 3.2 of \omega_d(Z). This ambiguity affects the constants in Theorem 4.2 and Theorem 4.3; the metric convention for B^n should be fixed from the start.
- [General] There are occasional wording issues, such as 'zeroes' for 'zeros' and the unnumbered equation (4.1); these are cosmetic but should be cleaned up in revision.
Circularity Check
Lower-bound half of the central equivalence is an immediate corollary of the same author's [18]; the upper-bound construction is independent, so the paper is only partially self-citation-dependent.
-
self citation load bearing
[Section 5, Proof of Theorem 4.1; Theorem 4.1 statement in Section 4]
"We obtain Theorem 4.1 as a corollary of a “Remez-type inequality for smooth functions”, obtained in [18]. ... In order to prove Theorem 4.1, it remains to notice that, since always Rd(Z) ≥ 1, we have RGd(Z) = inf f∈Ud(Z) Md+1(f) ≥ (d+1)!/(Rd(Z)+1) ≥ (d+1)!/(2Rd(Z)) = (d+1)!/2 \hat Rd(Z)."
Theorem 4.1, the lower-bound half of the advertised approximate equivalence, is not derived by new arguments in this paper. It is a direct corollary of the same author's [18, Theorem 5.2], which already states M_{d+1}(f) ≥ (d+1)!/(R_d(Z)+1) for every f vanishing on Z with M0(f)=1. Substituting the definition RG_d(Z) = inf_{f∈U_d(Z)} M_{d+1}(f) and using R_d(Z) ≥ 1 gives Theorem 4.1 in one line. Thus that half of the central claim reduces to a load-bearing self-citation rather than an independent first-principles derivation. The upper-bound direction (Theorem 4.2) is an independent construction, so the paper is only partially dependent on the self-citation chain.
full rationale
The main identifiable derivation that reduces to its own input is the lower-bound theorem: Theorem 4.1 is literally the same author's earlier [18, Theorem 5.2] restated with the definition of RG_d. This is self-citation load-bearing for one half of the claimed equivalence, but it is not full constructional circularity because [18] is a separately published result whose stated assumptions do not include the present theorem, and the upper-bound proof of Theorem 4.2 is a new bump-correction construction. The paper's proof of Theorem 4.2 does not verify that the corrected function f has M0(f)=1 before normalizing, so the upper-bound argument is incomplete at that step; however, that gap is a correctness issue, not a circularity. Similarly, Corollary 4.1 and Theorem 4.3 invoke earlier same-author Remez-type results [17,18], but those are used as external lemmas rather than as redefinitions of the target quantities. Overall, the finite-set equivalence has genuine independent content on the upper-bound side, while the lower-bound side is inherited from prior work by the same author.
Assumptions & free parameters
assumptions (5)
- standard math Polynomial interpolation with Lagrange remainder (Proposition 1.1) is valid for C^{d+1} functions on [-1,1].
- standard math Theorem 3.2 of [17]: the inverse Remez constant is bounded below in terms of omega_d(Z), normalized by m_n(B^n).
- standard math Theorem 5.2 of [18]: for a C^{d+1} function f vanishing on Z with M0(f)=1, M_{d+1}(f) >= (d+1)!/(R_d(Z)+1).
- standard math The Vitushkin polynomial bound M_{n,d}(epsilon) <= (4d)^n epsilon^{-(n-1)} holds for all epsilon > 0.
- standard math There exists a C-infinity bump psi on B^n with psi(0)=1, vanishing near the boundary, with bounded derivatives.
Cite this review
Pith. "Pith review of Smooth rigidity and Remez-type inequalities." pith.science (2026). https://pith.science/paper/R2YE3KMO
@misc{pith2026200913994,
author = {Pith},
title = {Pith review of: Smooth rigidity and Remez-type inequalities},
year = {2026},
howpublished = {\url{https://pith.science/paper/R2YE3KMO}},
note = {Machine review of arXiv:2009.13994}
}
abstract
If a smooth function of one variable has maximum one on the unit interval, and has there $d$ zeroes, then its $(d+1)$-st derivative must be "big". This is one of the simplest examples of what we call "smooth rigidity": certain geometric properties of zero sets of smooth functions $f$ imply explicit lower bounds on the high-order derivatives of $f$. In dimensions greater than one, the powerful one-dimension tools, like Lagrange's remainder formula, and divided finite differences, are not directly applicable. Still, the result above implies, via line sections, rather strong restrictions on zeroes of smooth functions of several variables \cite{Yom1}). In the present paper we study the geometry of zero sets of smooth functions, and significantly extend the results of \cite{Yom1}, including into consideration, in particular, finite zero sets (for which the line sections usually do not work). Our main goal is to develop a truly multi-dimensional approach to smooth rigidity, based on polynomial Remez-type inequalities (which compare the maxima of a polynomial on the unit ball, and on its subset). Very informally, one of our main results is that a "smooth rigidity" of a zeroes set $Z$ is approximately the "inverse Remez constant" of $Z$.
Reference graph
Works this paper leans on
-
[17]
Yomdin, Remez-Type Inequality for Discrete Sets, Isr
Y. Yomdin, Remez-Type Inequality for Discrete Sets, Isr. J. of Math. , Vol 186 (Nov. 2011), 45-60
work page 2011
-
[1]
Brudnyi, On Covering Numbers of Sublevels Sets of Analytic Fun c- tions, J
A. Brudnyi, On Covering Numbers of Sublevels Sets of Analytic Fun c- tions, J. of Appr. Theory , 162 (2010), no. 1, 72-93
work page 2010
-
[2]
A. Brudnyi, Y. Yomdin, Norming sets, and related Remez-type ine qual- ities, J. Aust. Math. Soc. 100 (2016) 163–181
work page 2016
-
[3]
Y. Brudnyi, M. Ganzburg, On an extremal problem for polynomials of n variables, Math. USSR Izv. 37 (1973), 344-355
work page 1973
-
[4]
Y. Brudnyi and P. Shvartsman, Generalizations of Whitneys ext ension theorem, IMRN 1994, no. 3, 129–139
work page 1994
-
[5]
Erdelyi, Remez-type inequalities and their applications, J
T. Erdelyi, Remez-type inequalities and their applications, J. Comp. Appl. Math. 47 (1993) 167-209
work page 1993
-
[6]
Favard, Sur l’interpolation, Bull
J. Favard, Sur l’interpolation, Bull. de la S. M. F. , 67 (1939), 103-113
work page 1939
-
[7]
Fefferman, Whitneys extension problem for C m, Ann
C. Fefferman, Whitneys extension problem for C m, Ann. of Math. 164 (2006), 313–359
work page 2006
Show all 18 references
-
[8]
Fefferman, B
C. Fefferman, B. Klartag, Fitting a C m-smooth function to data. Part I, Ann. of Math. (2) 169 (2009), 315346. Part II, Rev. Mat. Iberoam. 25 (2009), 49273
2009
-
[9]
Friedland, Y
O. Friedland, Y. Yomdin, Vitushkin-Type theorems, in Geometric As- pects of Functional Analysis , Springer, Cham, 2014, pp. 147–157
2014
-
[10]
E. J. Remez, Sur une propriete des polynomes de Tchebycheff, Comm. Inst. Sci. Kharkov 13 (1936) 93-95
1936
-
[11]
A. G. Vitushkin, O mnogomernyh Variaziyah, Gostehisdat, Mosk ow, (1955)
1955
-
[12]
Whitney, Analytic extensions of differentiable functions defin ed in closed sets, Trans
H. Whitney, Analytic extensions of differentiable functions defin ed in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63–89
1934
-
[13]
Whitney, Differentiable functions defined in closed sets
H. Whitney, Differentiable functions defined in closed sets. I, Tr ans. Amer. Math. Soc. 36 (1934), 369–387. 13
1934
-
[14]
Whitney, Functions differentiable on the boundaries of region s, Ann
H. Whitney, Functions differentiable on the boundaries of region s, Ann. of Math. 35 (1934), 482–485
1934
-
[15]
almost polynomial
Y. Yomdin, The set of zeroes of an “almost polynomial” function, Proc. AMS, Vol. 90, No. 4 (1984), 538-542
1984
-
[16]
Yomdin, G
Y. Yomdin, G. Comte, Tame geometry with application in smooth an al- ysis, Springer, LNM 1843, 2004
2004
-
[18]
Ge ometry and its applications
Y. Yomdin, Remez-Type Inequality for smooth functions, in “Ge ometry and its applications”, V. Rovenski, P. Walczak, Editors, Springer, C ham. 2014, 235-243. 14
2014
Reviewed August 27, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.