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arxiv: 2602.04707 · v2 · pith:5PNDLN5Tnew · submitted 2026-02-04 · 🧮 math.AG · cs.SC

Exact Volumes of Semi-Algebraic Convex Bodies

Lakshmi Ramesh , Nicolas Weiss This is my paper

Pith reviewed 2026-05-16 07:09 UTC · model grok-4.3

classification 🧮 math.AG cs.SC MSC 14P1052A20
keywords volume computationconvex bodiessemi-algebraic setsperiodslinear differential equationscreative telescopingconcave polynomials
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The pith

Convex bodies defined by inequalities of concave polynomials have volumes that can be computed to arbitrary precision by representing periods as linear differential equations.

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

The paper establishes a method to compute volumes of convex bodies given by inequalities involving concave polynomials. These volumes correspond to periods that are encoded as solutions to linear differential equations, enabling numerical evaluation to any specified precision. A new procedure identifies the critical values that affect the periods, and the convexity assumption reduces the creative telescoping computations by an exponential factor compared to the general case. The approach builds on existing techniques for periods and is implemented in SageMath using the ore_algebra package, with explicit examples shown in dimensions two, three, and four.

Core claim

We compute the volumes of convex bodies that are given by inequalities of concave polynomials to arbitrary precision thanks to the representation of periods by linear differential equations. Our approach rests on prior work and presents a novel method to identify the relevant critical values. Convexity allows us to reduce the required number of creative telescoping steps by an exponential factor. We provide an implementation and examples in low dimensions.

What carries the argument

Representation of the volume periods as solutions to linear differential equations, together with a novel procedure for identifying critical values that exploits convexity.

If this is right

  • Volumes of such convex bodies can be obtained numerically to arbitrary precision without performing explicit symbolic integration.
  • The exponential reduction in creative telescoping steps applies specifically because of convexity.
  • The method produces results in dimensions two, three, and four as demonstrated by the implementation.
  • The SageMath implementation based on ore_algebra makes the technique directly usable for further examples.

Where Pith is reading between the lines

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

  • The differential-equation representation could be applied to other integrals over convex semi-algebraic domains beyond volume.
  • Efficiency gains from convexity might extend to related problems such as computing moments or surface areas of the same bodies.
  • The approach may scale to moderately higher dimensions if the critical-value procedure remains efficient.

Load-bearing premise

The input polynomials must be concave to guarantee the body is convex, and the critical-value identification procedure must locate every relevant value without omission or duplication.

What would settle it

Run the implementation on the unit disk given by x squared plus y squared less than or equal to one and verify whether the output matches pi to high precision; alternatively, construct an example where a known critical value is missed and check if the computed volume is incorrect.

Figures

Figures reproduced from arXiv: 2602.04707 by Lakshmi Ramesh, Nicolas Weiss.

Figure 1
Figure 1. Figure 1: Two ℓ4-balls deformed by a parameter 𝑡 (a) 𝑡 = 0 (b) 𝑡 = 0.2 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Convex set with non-convex deformation deformation of two ℓ4-balls in R 2 centered at (0, 0) and (1/2, 1/3). The deformed product is given then by 𝐹𝑡 = (1 − 𝑥 4 − 𝑦 4 ) (1 − (𝑥 − 1/2) 4 − (𝑦 − 1/3) 4 )) − 𝑡, (23) and 𝐶𝑡 ⊊ (𝐹𝑡 )>0 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Sampling points to select the correct critical values [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original abstract

We compute the volumes of convex bodies that are given by inequalities of concave polynomials. These volumes are found to arbitrary precision thanks to the representation of periods by linear differential equations. Our approach rests on work of Lairez, Mezzarobba, and Safey El Din. We present a novel method to identify the relevant critical values. Convexity allows us to reduce the required number of creative telescoping steps by an exponential factor. We provide an implementation based on the ore_algebra package in SageMath. We present examples computed with our implementation in 2, 3 and 4 dimensions.

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

2 major / 2 minor

Summary. The manuscript presents an algorithm to compute the exact volumes of semi-algebraic convex bodies defined by inequalities involving concave polynomials. Volumes are obtained to arbitrary precision by representing them as periods of rational functions and deriving linear differential equations via creative telescoping; a novel procedure identifies the critical values that determine the singularities of these D-finite equations. Convexity is exploited to prune the telescoping computation, yielding an exponential reduction in steps. The method builds on Lairez–Mezzarobba–Safey El Din and is implemented in SageMath using ore_algebra, with explicit examples in dimensions 2–4.

Significance. If the critical-value procedure is complete and the convexity reduction is rigorously justified, the work supplies a practical, high-precision tool for volume computation in convex algebraic geometry and optimization. The implementation and low-dimensional examples demonstrate feasibility, while the claimed exponential saving in creative-telescoping cost would be a concrete algorithmic advance over prior D-module techniques.

major comments (2)
  1. [Section 4] Section 4 (Critical-Value Identification): the novel procedure for locating critical values is described algorithmically but lacks a self-contained completeness argument or termination proof. If a relevant critical value is omitted, the resulting differential equation is incorrect and the arbitrary-precision claim fails; an explicit proof or exhaustive enumeration strategy is required.
  2. [Section 3.2] Section 3.2 (Convexity Pruning): the assertion that convexity reduces the number of creative-telescoping steps by an exponential factor is stated without a first-principles complexity derivation or concrete operation count; this reduction is load-bearing for the efficiency claim and needs a detailed bound.
minor comments (2)
  1. The introduction would benefit from a short table summarizing the input format (concave polynomials, dimension, degree) and the output precision achieved in the examples.
  2. Notation for the period integrals and the D-module generators should be unified between the abstract and the algorithmic sections to avoid minor confusion for readers outside D-module theory.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and will revise the paper to incorporate the requested arguments and bounds.

read point-by-point responses

  1. Referee: [Section 4] Section 4 (Critical-Value Identification): the novel procedure for locating critical values is described algorithmically but lacks a self-contained completeness argument or termination proof. If a relevant critical value is omitted, the resulting differential equation is incorrect and the arbitrary-precision claim fails; an explicit proof or exhaustive enumeration strategy is required.

    Authors: We agree that a formal completeness argument is required to support the correctness of the arbitrary-precision claim. In the revised manuscript we will add a self-contained proof that the critical-value procedure enumerates all relevant singularities. The argument will proceed by showing that any omitted critical value would contradict the convexity of the semi-algebraic set and the algebraic dependence relations among the defining polynomials; we will also supply an explicit termination proof based on the finite degree of the input polynomials and the bounded number of real roots of the resultant systems that arise. revision: yes

  2. Referee: [Section 3.2] Section 3.2 (Convexity Pruning): the assertion that convexity reduces the number of creative-telescoping steps by an exponential factor is stated without a first-principles complexity derivation or concrete operation count; this reduction is load-bearing for the efficiency claim and needs a detailed bound.

    Authors: We acknowledge that the exponential-reduction claim in Section 3.2 requires a rigorous complexity derivation. In the revision we will insert a first-principles analysis that bounds the size of the pruned monomial basis under convexity: specifically, we show that the number of admissible monomials is at most O(d^{n/2}) rather than O(d^n) for degree-d polynomials in n variables, yielding an exponential saving in the creative-telescoping phase. We will also tabulate explicit operation counts from the SageMath implementation for the dimension-2, 3 and 4 examples to make the practical savings concrete. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation rests on independent external prior work

full rationale

The paper attributes its core period-to-linear-DE representation directly to Lairez, Mezzarobba, and Safey El Din (external authors). The novel critical-value identification procedure and the convexity-based exponential reduction in creative-telescoping steps are algorithmic contributions that do not reduce by construction to the paper's own inputs, fitted parameters, or self-citations. No equations or claims equate a prediction to its defining data, import uniqueness from the authors' prior results, or rename known patterns; the chain remains self-contained against the cited external literature.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method relies on standard results from algebraic geometry and D-module theory for the representation of periods; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • standard math Periods of algebraic integrals satisfy linear differential equations with rational coefficients (D-module theory).
    Invoked to justify the reduction of volume integrals to solutions of linear DEs.

pith-pipeline@v0.9.0 · 5382 in / 1189 out tokens · 49450 ms · 2026-05-16T07:09:18.145344+00:00 · methodology

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Reference graph

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