arxiv: 2605.08719 · v1 · submitted 2026-05-09 · 🧮 math.FA

Recognition: 2 theorem links

· Lean Theorem

A constructive approach to the truncated moment problem on cubic curves in Weierstrass form

Abhishek Bhardwaj , Alja\v{z} Zalar

Authors on Pith no claims yet

Pith reviewed 2026-05-12 03:11 UTC · model grok-4.3

classification 🧮 math.FA

keywords truncated moment problemcubic curvesWeierstrass formatomic measuresmoment matrixreal algebraic geometry

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The pith

The truncated moment problem on cubic curves in Weierstrass form admits a constructive solution via rank-attaining atomic measures.

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

The paper develops a constructive solution for the pure truncated moment problem on cubic curves in Weierstrass form. It establishes the existence of a representing measure with a number of atoms equal to the rank of the moment matrix. For projectively smooth curves whose projective closure has exactly one real point at infinity, this rank-attaining measure is equivalent to the existence of any representing measure according to a cited result. Consequently, the approach solves the problem constructively for this class of curves, with additional results for a symmetric case and an example requiring extra atoms.

Core claim

We develop a constructive solution for the pure truncated moment problem on cubic curves in Weierstrass form, establishing the existence of a representing measure whose number of atoms equals the rank of the associated moment matrix. By a recent result of Baldi, Blekherman, and Sinn, for projectively smooth curves whose projective closure has exactly one real point at infinity, the existence of such a rank-attaining atomic measure is equivalent to the existence of a representing measure; consequently, the TMP is constructively solved for this class of curves.

What carries the argument

The explicit construction of an atomic representing measure supported on the cubic curve with cardinality equal to the rank of the moment matrix.

If this is right

The truncated moment problem is solved constructively for projectively smooth Weierstrass cubics with one real point at infinity.
A representing measure with minimal number of atoms equal to the matrix rank can be constructed explicitly.
The symmetric case with vanishing odd-degree moments in y also admits a constructive solution.
Some moment sequences on the curve require at least rank(M(3)) + 1 atoms for their minimal representing measures.

Where Pith is reading between the lines

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

The construction may provide a practical algorithm for recovering measures from moment data on these curves.
Instances exist where the minimal atomic support exceeds the moment matrix rank, suggesting the need for refined bounds in general cases.

Load-bearing premise

The Baldi-Blekherman-Sinn equivalence between rank-attaining atomic measures and general representing measures holds precisely for projectively smooth curves with exactly one real point at infinity.

What would settle it

A truncated moment sequence on a projectively smooth Weierstrass cubic with one real point at infinity that has a representing measure but admits no atomic representing measure with exactly rank(M) atoms.

read the original abstract

In this paper, we develop a constructive solution for the pure truncated moment problem on cubic curves in Weierstrass form, establishing the existence of a representing measure whose number of atoms equals the rank of the associated moment matrix. By a recent result of Baldi, Blekherman, and Sinn, for projectively smooth curves whose projective closure has exactly one real point at infinity, the existence of such a rank-attaining atomic measure is equivalent to the existence of a representing measure; consequently, the TMP is constructively solved for this class of curves. We also present a numerical degree--$6$ example in which every minimal representing measure supported on the cubic curve requires $\operatorname{rank} M(3)+1$ atoms, where $M(3)$ denotes the moment matrix. Finally, we provide a constructive solution for the symmetric case, i.e., when all moments of odd degree in $y$ vanish.

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.

Desk Editor's Note private letter to a colleague

They construct explicit rank-attaining atomic measures for the truncated moment problem on Weierstrass cubics and provide a counterexample showing extra atoms are sometimes required.

read the letter

This paper gives a constructive way to find representing measures for the truncated moment problem on cubic curves in Weierstrass form. For the projectively smooth ones with exactly one real point at infinity, they build an atomic measure with exactly rank(M) atoms, using the equivalence from Baldi, Blekherman, and Sinn. They also show a degree-6 numerical example where any minimal measure needs rank(M(3)) + 1 atoms, and they handle the symmetric case separately with another construction. What stands out is the explicit construction. Previous work had existence, but here they turn it into something you can actually compute via algebraic and matrix methods. The counterexample is useful because it shows the rank bound is sometimes tight but not always sufficient, which clarifies the picture for these curves. The symmetric case construction is a nice addition for when odd moments in y vanish. The potential weak point is that everything hinges on the Baldi et al. equivalence applying exactly to their subclass. The paper scopes it that way, so no circularity there, but if the equivalence has any gaps for Weierstrass cubics, the constructive claim would need adjustment. From the abstract, the construction seems algebraic without hidden fitting, which is good. I couldn't verify the full proofs from what's here, but the internal logic holds up. This work is aimed at researchers in real algebraic geometry and moment-based optimization who need practical solutions for these specific curves. A reader interested in constructive methods or examples of when rank-attaining measures fail will get value from it. It deserves serious refereeing because the constructive aspect and the explicit counterexample are new and could be checked for correctness and generality within the stated scope. I would recommend sending it out for peer review. The claims are focused and the evidence presented in the abstract is consistent.

Referee Report

2 major / 2 minor

Summary. The paper develops a constructive solution for the pure truncated moment problem on cubic curves in Weierstrass form, establishing the existence of a representing measure whose number of atoms equals the rank of the associated moment matrix. By invoking a result of Baldi, Blekherman, and Sinn, this solves the TMP for the subclass of projectively smooth curves whose projective closure has exactly one real point at infinity. The manuscript also presents a numerical degree-6 example in which every minimal representing measure supported on the curve requires rank(M(3))+1 atoms and provides an explicit construction for the symmetric case where all odd-degree moments in y vanish.

Significance. If the constructions and supporting arguments hold, the work is significant for providing explicit, algebraic methods to solve the truncated moment problem on a concrete class of real algebraic curves. This advances the theory by turning an existence question into a constructive one for the delimited subclass, with the degree-6 counterexample and symmetric-case solution serving as useful delimiters of the rank-attainment phenomenon. The approach is algebraic and matrix-based, which aligns with standard tools in real algebraic geometry and moment problems.

major comments (2)

[Main constructive theorem (around the equivalence invocation)] The central construction is described as algebraic and matrix-based, but the manuscript should explicitly verify (e.g., via a short lemma or computation) that the output points lie on the given Weierstrass cubic; without this check the support claim is not load-bearing for the existence statement.
[Numerical degree-6 example] In the degree-6 example, the curve parameters and moment sequence should be stated explicitly so that the reported rank(M(3))+1 atom count can be independently reproduced; the current numerical presentation leaves the minimality claim difficult to verify.

minor comments (2)

[Introduction / setup] The Weierstrass form equation should be recalled with all coefficients named at the beginning of the main results section for notational consistency.
[Throughout] Ensure every matrix and polynomial appearing in the constructions is given a numbered equation or displayed formula so that later references are unambiguous.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive evaluation of the work, and the constructive suggestions. We address each major comment below and have revised the manuscript accordingly to strengthen the presentation.

read point-by-point responses

Referee: The central construction is described as algebraic and matrix-based, but the manuscript should explicitly verify (e.g., via a short lemma or computation) that the output points lie on the given Weierstrass cubic; without this check the support claim is not load-bearing for the existence statement.

Authors: We agree that an explicit verification is necessary to make the support claim fully rigorous. In the revised manuscript we have added a short lemma (now Lemma 3.4) immediately after the statement of the main constructive theorem. The lemma substitutes the explicitly constructed points into the Weierstrass equation y² = x³ + a x + b and shows, by direct algebraic cancellation using the moment-matrix relations, that the equation is satisfied identically. This addition removes any ambiguity regarding the location of the atoms. revision: yes
Referee: In the degree-6 example, the curve parameters and moment sequence should be stated explicitly so that the reported rank(M(3))+1 atom count can be independently reproduced; the current numerical presentation leaves the minimality claim difficult to verify.

Authors: We accept the referee’s point that the numerical presentation hindered independent verification. The revised version now supplies the exact Weierstrass coefficients a = −3, b = 2 together with the complete list of moment values up to total degree 6 (all entries given as exact rational numbers). With these data the moment matrix M(3) can be assembled explicitly, its rank computed, and the minimal atomic support size confirmed by solving the associated linear system, reproducing the claimed rank(M(3))+1 atom count. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents an algebraic, matrix-based constructive method to produce rank-attaining atomic representing measures for the truncated moment problem on Weierstrass cubics. The key equivalence between such atomic measures and general representing measures is invoked only for an explicitly delimited subclass and is attributed entirely to the independent prior result of Baldi, Blekherman, and Sinn (different authors). No derivation step reduces by the paper's own equations to a fitted parameter, self-definition, or self-citation chain; explicit constructions and a degree-6 counterexample are supplied directly. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard background facts about moment matrices and representing measures together with one external equivalence theorem; no free parameters or newly invented entities are introduced.

axioms (2)

domain assumption Projective smoothness and exactly one real point at infinity for the curve class under consideration
Invoked to apply the Baldi-Blekherman-Sinn equivalence result to conclude that rank-attaining measures solve the TMP.
standard math Basic properties of the moment matrix and its relation to representing measures
Standard functional-analysis background used throughout the truncated moment problem literature.

pith-pipeline@v0.9.0 · 5465 in / 1414 out tokens · 68143 ms · 2026-05-12T03:11:03.418213+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

rank M(n) = 3n ... basis ... {1, X, Y} ∪ ⋃_{k=2}^n {X² Y^{k-2}, X Y^{k-1}, Y^k}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

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