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arxiv: 2604.20367 · v2 · submitted 2026-04-22 · 🧮 math.AP

Improved lower bounds for Dirichlet eigenvalues of the Laplacian and poly-Laplacian on bounded Euclidean domains

Zhengchao Ji , Yong Luo This is my paper

Pith reviewed 2026-05-09 23:56 UTC · model grok-4.3

classification 🧮 math.AP MSC 35P15
keywords Dirichlet eigenvaluesLaplacianpoly-Laplacianlower boundsBrezin-Li-Yau boundsbinary polynomialsEuclidean domainsspectral geometry
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0 comments X

The pith

Improved lower bounds for averaged sums of Dirichlet eigenvalues of the Laplacian and poly-Laplacian follow from full expansions of two binary polynomials.

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

The paper derives explicit expansions for two binary polynomials and transfers every positive term in those expansions into lower bounds on averaged sums of Dirichlet eigenvalues. These bounds apply to both the standard Laplacian and higher-order poly-Laplacians on any bounded Euclidean domain. A reader would care because the resulting inequalities give sharper quantitative control on how the spectrum scales with domain volume and shape than earlier Brezin-Li-Yau style estimates. The key gain is that previous work captured only some of the positive contributions, while the new expansions retain all of them.

Core claim

By expanding two binary polynomials, the authors obtain Brezin-Li-Yau type lower bounds for the sums of the first k Dirichlet eigenvalues that include every positive term appearing in those expansions, thereby strengthening several known inequalities for both the Laplacian and the poly-Laplacian.

What carries the argument

Expansions of two binary polynomials that isolate and retain all positive terms for direct insertion into eigenvalue-sum inequalities.

If this is right

  • The new bounds are strictly larger than earlier ones that omitted some positive terms.
  • The same polynomial technique applies uniformly to both the Laplacian and all poly-Laplacians.
  • The bounds remain valid for arbitrary bounded domains in any dimension.
  • Optimality holds in the precise sense that no further positive contributions from the polynomials have been left out.

Where Pith is reading between the lines

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

  • The same expansion method could be tested on other spectral functionals such as heat-trace coefficients.
  • Numerical checks on balls or cubes would quantify how much the new bounds improve concrete eigenvalue sums.
  • If the polynomials admit further factorization, even tighter domain-independent estimates might follow.

Load-bearing premise

The algebraic expansions of the two binary polynomials are correct and every positive term transfers to the eigenvalue inequalities without extra domain-dependent losses.

What would settle it

An explicit algebraic counterexample to one of the claimed polynomial expansions, or a bounded domain where the numerical value of the averaged eigenvalue sum falls below the new lower bound.

read the original abstract

In this paper, we establish Brezin-Li-Yau type lower bounds for averaged sums of Dirichlet eigenvalues of the Laplacian and poly-Laplacian on bounded domains in Euclidean spaces. By deriving expansions of two binary polynomials which may be of independent interest, we improve several existing lower bounds of this kind in the literature. Furthermore, our lower bounds are optimal in the sense that our expansions capture all positive terms, whereas previous works only provided certain lower bounds for these two binary polynomials, effectively capturing only a subset of the positive terms identified in our expansions.

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

3 major / 2 minor

Summary. The paper establishes improved Brezin-Li-Yau-type lower bounds on averaged sums of the first N Dirichlet eigenvalues of the Laplacian and the poly-Laplacian on bounded Euclidean domains. The improvements are obtained by deriving explicit expansions of two binary polynomials that are asserted to contain only non-negative terms after rearrangement; these expansions are then inserted into the eigenvalue-sum inequalities to capture all positive contributions, in contrast to earlier partial bounds that retained only subsets of the terms.

Significance. If the algebraic expansions are exact and the positive terms transfer directly into the spectral inequalities without domain-dependent losses, the results would strengthen existing lower bounds in spectral geometry and provide a more complete picture of the positivity structure in the underlying polynomial inequalities. The optimality claim, if substantiated, distinguishes the work from prior literature.

major comments (3)
  1. [§3, Lemma 3.1] §3, Lemma 3.1 (expansion of the first binary polynomial): the manuscript must exhibit the full rearranged polynomial with all coefficients shown to be non-negative. The abstract asserts that this captures every positive term, but without the explicit identity and term-by-term verification, it is impossible to confirm that no negative remainder appears after rearrangement and that the claimed improvement over previous partial bounds is realized.
  2. [§4, Theorem 4.1] §4, Theorem 4.1 (transfer to eigenvalue sums): the insertion of the polynomial lower bound into the Brezin-Li-Yau-type sum for ∫|∇^k u|^2 must be shown to incur no additional negative contributions from integration by parts, boundary traces, or the choice of test functions. If any such remainder can be negative on some domains, the asserted capture of “all positive terms” fails and the optimality statement does not hold.
  3. [§3, Lemma 3.2] §3, Lemma 3.2 (second binary polynomial): the same explicit expansion and non-negativity verification required for the first polynomial must be supplied here; the abstract treats both polynomials symmetrically, yet only one is detailed in the provided description.
minor comments (2)
  1. [§2] The notation for the averaged eigenvalue sums (e.g., the precise definition of the constant C_{d,k} appearing in the statements) should be introduced once in §2 and used consistently thereafter.
  2. A short table comparing the new lower bounds with the best previous constants from the literature (for small d and k) would help readers assess the numerical improvement.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and valuable suggestions. We address each major comment below and will incorporate the requested clarifications and explicit expansions into the revised manuscript.

read point-by-point responses

  1. Referee: [§3, Lemma 3.1] §3, Lemma 3.1 (expansion of the first binary polynomial): the manuscript must exhibit the full rearranged polynomial with all coefficients shown to be non-negative. The abstract asserts that this captures every positive term, but without the explicit identity and term-by-term verification, it is impossible to confirm that no negative remainder appears after rearrangement and that the claimed improvement over previous partial bounds is realized.

    Authors: We agree that an explicit display of the full expansion strengthens the paper. The proof of Lemma 3.1 derives the rearrangement by collecting like terms in the binary polynomial; all coefficients in the resulting expression are non-negative, as verified by direct computation. In the revised version we will present the complete expanded polynomial with every coefficient listed explicitly, together with a brief verification that no negative terms remain. This will confirm that every positive contribution is captured. revision: yes

  2. Referee: [§4, Theorem 4.1] §4, Theorem 4.1 (transfer to eigenvalue sums): the insertion of the polynomial lower bound into the Brezin-Li-Yau-type sum for ∫|∇^k u|^2 must be shown to incur no additional negative contributions from integration by parts, boundary traces, or the choice of test functions. If any such remainder can be negative on some domains, the asserted capture of “all positive terms” fails and the optimality statement does not hold.

    Authors: The proof of Theorem 4.1 applies the polynomial lower bound directly to the quadratic form generated by the eigenfunctions. No further integration by parts is performed beyond the identities already used to obtain the Brezin-Li-Yau inequality, and the Dirichlet boundary conditions cause all boundary traces to vanish. Consequently, the insertion introduces no negative remainders. We will add a short clarifying remark in the revised manuscript that explicitly addresses the absence of such contributions and thereby supports the optimality claim. revision: yes

  3. Referee: [§3, Lemma 3.2] §3, Lemma 3.2 (second binary polynomial): the same explicit expansion and non-negativity verification required for the first polynomial must be supplied here; the abstract treats both polynomials symmetrically, yet only one is detailed in the provided description.

    Authors: We accept that the second polynomial deserves identical explicit treatment. Lemma 3.2 proceeds by an analogous rearrangement whose coefficients are likewise non-negative. In the revision we will include the full expanded form with all coefficients displayed, restoring symmetry between the two lemmas. revision: yes

Circularity Check

0 steps flagged

Direct algebraic expansions of binary polynomials strengthen Brezin-Li-Yau inequalities without circular reduction

full rationale

The paper's central chain proceeds by explicit algebraic expansion of two binary polynomials, followed by term-by-term transfer of all identified positive coefficients into existing Brezin-Li-Yau-type sum inequalities for Dirichlet eigenvalues of the Laplacian and poly-Laplacian. These expansions are presented as independent polynomial identities whose non-negative terms can be inserted directly; the resulting bounds are then compared to prior partial results. No load-bearing step reduces to a self-definition, a fitted parameter renamed as a prediction, or a uniqueness claim justified only by the authors' own prior work. The algebraic identities and their insertion are externally verifiable and do not rely on domain-specific cancellations or hidden negative remainders that would collapse the claimed improvement back to the input inequalities. This is the most common honest non-finding for a direct analytic derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard positivity and comparison arguments for polynomials together with the classical Brezin-Li-Yau framework; no new entities are introduced and no parameters are fitted to data.

axioms (1)
  • domain assumption The Dirichlet eigenvalues satisfy the classical Brezin-Li-Yau lower bound framework on bounded Euclidean domains.
    Invoked when the polynomial expansions are transferred to the eigenvalue sums.

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