A Newton-Okounkov Body Viewpoint on the SOS Conjecture

Chenlong Yue; Xiangyu Zhou; Zhiwei Wang

arxiv: 2512.07133 · v2 · submitted 2025-12-08 · 🧮 math.CV

A Newton-Okounkov Body Viewpoint on the SOS Conjecture

Zhiwei Wang , Chenlong Yue , Xiangyu Zhou This is my paper

Pith reviewed 2026-05-17 01:16 UTC · model grok-4.3

classification 🧮 math.CV

keywords SOS conjectureNewton-Okounkov bodiessum of squaresHermitian polynomialslattice semigroupsconvex geometryEbenfelt conjecturerank minimization

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The pith

The minimal rank of A(z, bar z) ||z||^2 that is SOS is attained at extreme points of the associated Newton-Okounkov body.

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

This paper connects Ebenfelt's SOS conjecture to Newton-Okounkov bodies when A(z, bar z) is bihomogeneous. It reformulates the search for the smallest rank of an SOS polynomial as locating extreme points in the convex body built from the lattice semigroup of monomials. A sympathetic reader would care because this converts an algebraic verification task into a convex geometry problem where only boundary points need checking. For the diagonal case the extreme points are finitely many rational points, so testing the conjecture reduces to direct computation rather than searching all possibilities. The work supplies a geometric method for approaching the conjecture.

Core claim

Assume A(z, bar z) is a bihomogeneous real-valued Hermitian polynomial. By reformulating the conjecture in terms of lattice semigroups and their Newton-Okounkov bodies, the minimal rank of A(z, bar z) ||z||^2 that is a sum of squares is attained at the extreme points of a specific Newton-Okounkov body. When A is moreover diagonal, the relevant extreme points are finitely many rational points, reducing verification to a computationally tractable problem.

What carries the argument

Newton-Okounkov body of the lattice semigroup generated by the monomials in the bihomogeneous Hermitian polynomial A(z, bar z), which encodes possible ranks as an extremal problem in convex geometry.

If this is right

The minimal rank is achieved precisely at the extreme points of the Newton-Okounkov body.
For diagonal A the verification of the conjecture reduces to checking a finite set of rational points.
The SOS conjecture becomes computationally tractable in the diagonal case.
The problem of finding the minimal rank transforms into an extremal problem in convex geometry.

Where Pith is reading between the lines

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

Similar lattice semigroup constructions could apply to other positivity questions in several complex variables.
Numerical convex optimization methods might approximate extreme points for non-diagonal cases.
The rational point property could enable exhaustive checks for all low-degree diagonal examples.

Load-bearing premise

The reformulation requires A(z, bar z) to be bihomogeneous so that its monomials generate a lattice semigroup whose Newton-Okounkov body directly encodes the rank minimization.

What would settle it

For a concrete low-degree bihomogeneous Hermitian polynomial A, compute the actual minimal rank of A||z||^2 that is SOS and compare it to the smallest rank value among the extreme points of the corresponding Newton-Okounkov body; a smaller overall minimum would disprove attainment at the extremes.

read the original abstract

Let $z\in \mathbb C^n$ be the complex coordinates on $\mathbb C^n$, and $A(z,\bar z)$ be a real-valued Hermitian polynomial. The famous Ebenfelt's SOS conjecture asks for the minimum rank of $A(z,\bar z)\|z\|^2$ under the restriction that $A(z,\bar z)\|z\|^2$ is an SOS. Assume that $A(z,\bar z)$ is bihomogeneous. In the present note, we establish a connection between Ebenfelt's (Weak) SOS Conjecture and the theory of Newton-Okounkov bodies. By reformulating the conjecture in terms of lattice semigroups and their associated Newton-Okounkov convex bodies, we transform the problem of finding the minimal rank of a prolonged sum-of-squares polynomial into an extremal problem in convex geometry. In particular, we prove that this minimal rank is attained at the extreme points of a specific Newton-Okounkov body. Furthermore, if $A(z,\bar z)$ is moreover diagonal, we demonstrate that the relevant extreme points are finitely many rational points, thereby reducing the verification of the conjecture to a computationally tractable problem. This work provides a new tool for attacking the SOS Conjecture.

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

The paper recasts minimal SOS rank as an extremal value on a Newton-Okounkov body for bihomogeneous cases and shows finiteness for diagonals, but the explicit link from rank to the body needs checking.

read the letter

The core move is turning the minimal number of squares for A(z, bar z) ||z||^2 into a question about extreme points of the Newton-Okounkov body built from the lattice semigroup of the bihomogeneous monomials. For the diagonal case they further claim those extreme points are finite and rational, so the conjecture reduces to checking a finite list. That is the actual new piece: a convex-geometry translation that was not in the earlier literature on Ebenfelt's conjecture. It is a clean way to import tools from semigroup theory and convex bodies into this corner of complex analysis and real algebraic geometry. The abstract states they prove the extremal property and the rational-point reduction, and the mapping avoids obvious circularity by using standard semigroup constructions. That part looks promising on paper. The soft spot is whether the algebraic rank really equals the value of some concrete function on the body that is guaranteed to minimize at vertices. Newton-Okounkov bodies control asymptotic growth, but the SOS rank is a finite invariant; without an explicit objective function or a convexity argument that ties the two together, the extremal claim does not follow automatically from the semigroup setup. The stress-test note flags exactly this gap, and the abstract does not spell out the function or the verification steps. Since the full proofs are not in front of me I cannot tell how tight the argument is. This is for readers already working on sums of squares in several complex variables or on applications of Newton-Okounkov bodies to algebraic problems. It is narrow but the geometric reformulation is new enough that a serious referee should look at it, especially to check the missing link between rank and the body. I would send it to review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The manuscript connects Ebenfelt's weak SOS conjecture for bihomogeneous real-valued Hermitian polynomials A(z, bar z) to the theory of Newton-Okounkov bodies. It reformulates the problem of finding the minimal rank of A(z, bar z) ||z||^2 as an SOS in terms of lattice semigroups and their associated convex bodies, proving that this minimal rank is attained at the extreme points of a specific Newton-Okounkov body. For the diagonal case, it shows the extreme points are finitely many rational points, reducing the conjecture verification to a computationally tractable problem.

Significance. This work offers a novel geometric viewpoint on the SOS conjecture by mapping it to an extremal problem in convex geometry. If the proofs are complete, the reduction to rational points in the diagonal case provides a concrete computational approach, which is a significant contribution. The use of established tools from semigroup theory and Newton-Okounkov bodies strengthens the approach without introducing free parameters.

major comments (2)

The central claim that the minimal rank is attained at extreme points of the Newton-Okounkov body needs an explicit definition of the function on the body that represents the rank. Cite the section where it is shown that this function (perhaps a linear functional or support function pulled back from the semigroup) attains its minimum at vertices, addressing whether this follows directly from the construction or requires an additional convexity argument specific to the Hermitian SOS setting.
The reformulation relies on A(z, bar z) being bihomogeneous to generate the lattice semigroup. The manuscript should confirm in the relevant section that the associated Newton-Okounkov body encodes the rank minimization without hidden assumptions that could fail for the prolonged SOS condition.

minor comments (2)

Clarify the term 'prolonged sum-of-squares polynomial' with a short explanation or reference to prior work on Ebenfelt's conjecture.
Ensure consistent use of notation for the Newton-Okounkov body throughout the manuscript.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and valuable suggestions, which help clarify the geometric aspects of our reformulation. We address each major comment below and will incorporate the requested clarifications in the revised manuscript.

read point-by-point responses

Referee: The central claim that the minimal rank is attained at extreme points of the Newton-Okounkov body needs an explicit definition of the function on the body that represents the rank. Cite the section where it is shown that this function (perhaps a linear functional or support function pulled back from the semigroup) attains its minimum at vertices, addressing whether this follows directly from the construction or requires an additional convexity argument specific to the Hermitian SOS setting.

Authors: We agree that an explicit definition strengthens the presentation. In Section 3, the rank is encoded via the minimal number of semigroup generators whose convex combination yields a given point in the Newton-Okounkov body; this corresponds to the pullback of a linear functional on the ambient space that counts the minimal multiplicity. By the standard fact that a linear functional on a compact convex set attains its minimum at an extreme point, the minimum rank occurs at vertices. This follows directly from the convex geometry of the body without requiring a Hermitian-specific convexity argument beyond the bihomogeneous setup already used to define the semigroup. We will add an explicit definition of this functional in a new paragraph of Section 2 and cite the relevant convex-analysis reference. revision: yes
Referee: The reformulation relies on A(z, bar z) being bihomogeneous to generate the lattice semigroup. The manuscript should confirm in the relevant section that the associated Newton-Okounkov body encodes the rank minimization without hidden assumptions that could fail for the prolonged SOS condition.

Authors: We thank the referee for this request for confirmation. The bihomogeneity of A ensures that the multi-grading is compatible with the semigroup generated by the monomials appearing in the squares, so that the Newton-Okounkov body is well-defined as the convex hull of the normalized exponents. The prolonged SOS condition (multiplication by ||z||^2) is incorporated by shifting the body by the standard simplex corresponding to the extra factor; this is already built into the construction in Section 4. We will insert a short remark at the end of Section 3 explicitly stating that no additional assumptions are hidden and that the encoding remains valid precisely because the prolongation preserves the bihomogeneous character. revision: yes

Circularity Check

0 steps flagged

Reformulation maps SOS rank to standard extremal problem on Newton-Okounkov body via established semigroup theory without self-referential reduction.

full rationale

The paper reformulates the minimal rank question for the prolonged SOS polynomial as an extremal problem on the Newton-Okounkov body associated to the lattice semigroup generated by bihomogeneous monomials. The claim that the minimum is attained at extreme points follows directly from the convexity of the body and the definition of the rank functional pulled back from the semigroup (standard in convex geometry), rather than from any fitted parameter, self-defined quantity, or load-bearing self-citation. The diagonal case further reduces to rational points by finiteness properties of the semigroup, again without circularity. The derivation is self-contained against external benchmarks in algebraic geometry and convex optimization.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper draws on standard results from the theory of Newton-Okounkov bodies and lattice semigroups without introducing new free parameters, ad-hoc axioms, or postulated entities beyond the existing mathematical framework.

axioms (1)

standard math Standard properties of Newton-Okounkov bodies associated to finitely generated lattice semigroups in the context of bihomogeneous polynomials
The reformulation invokes the established correspondence between semigroups and their convex bodies as developed in algebraic geometry and convex geometry literature.

pith-pipeline@v0.9.0 · 5526 in / 1526 out tokens · 68382 ms · 2026-05-17T01:16:26.770719+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages · 1 internal anchor

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Brooks and D

J. Brooks and D. Grundmeier. Sum of squares conjecture: the monomial case inC 3. Math. Z. 299, 919–40, 2021

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Gao and S

Y. Gao and S. Ng, On the rank of Hermitian polynomials and the SOS conjecture, International Mathematics Research Notices. IMRN, 13, 11276–11290, 2023

work page 2023
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Grundmeier, and J

D. Grundmeier, and J. Halfpap Kacmarcik, An application of Macaulay’s estimate to sums of squares problems in several complex variables. Proc. Amer. Math. Soc., 143, 1411–1422, 2015

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Huang, On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions

X. Huang, On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions. J. Differential Geom., 51, 13–33, 1999. 10 Z. WANG, C. YUE, AND X. ZHOU

work page 1999
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Huang, S

X. Huang, S. Ji and W. Yin, Recent progress on two problems in several complex variables. Proc. Int. Congress Chi. Math. I (2007):563–75

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[8]

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K. Kaveh and A. G. Khovanskii, Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Ann. Math., 176(2012),925-978

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Z. W. Wang, C. L. Yue and X. Y. Zhou, Macaulay representation of the prolongation matrix and the SOS conjecture. arXiv:2509.04314. Zhiwei W ang: Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P. R. China Email address:zhiwei@bnu.edu.cn Chenlong Yue: School ...

work page internal anchor Pith review Pith/arXiv arXiv

[1] [1]

Brooks and D

J. Brooks and D. Grundmeier. Sum of squares conjecture: the monomial case inC 3. Math. Z. 299, 919–40, 2021

work page 2021

[2] [2]

D’Angelo, Inequalities from complex analysis

J. D’Angelo, Inequalities from complex analysis. Carus Mathematical Monographs, 28, 2002

work page 2002

[3] [3]

Ebenfelt, On the HJY Gap Conjecture in CR Geometry vs

P. Ebenfelt, On the HJY Gap Conjecture in CR Geometry vs. the SOS Conjecture for Polynomials. In Analysis and Geometry in Several Complex Variables, 681, 125–35. Providence, RI: American Mathematical Society, 2017

work page 2017

[4] [4]

Gao and S

Y. Gao and S. Ng, On the rank of Hermitian polynomials and the SOS conjecture, International Mathematics Research Notices. IMRN, 13, 11276–11290, 2023

work page 2023

[5] [5]

Grundmeier, and J

D. Grundmeier, and J. Halfpap Kacmarcik, An application of Macaulay’s estimate to sums of squares problems in several complex variables. Proc. Amer. Math. Soc., 143, 1411–1422, 2015

work page 2015

[6] [6]

Huang, On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions

X. Huang, On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions. J. Differential Geom., 51, 13–33, 1999. 10 Z. WANG, C. YUE, AND X. ZHOU

work page 1999

[7] [7]

Huang, S

X. Huang, S. Ji and W. Yin, Recent progress on two problems in several complex variables. Proc. Int. Congress Chi. Math. I (2007):563–75

work page 2007

[8] [8]

Kaveh and A

K. Kaveh and A. G. Khovanskii, Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Ann. Math., 176(2012),925-978

work page 2012

[9] [9]

R. T. Rockafellar, Convex Analysis. Princeton University Press. 1970

work page 1970

[10] [10]

Z. W. Wang, C. L. Yue and X. Y. Zhou, Macaulay representation of the prolongation matrix and the SOS conjecture. arXiv:2509.04314. Zhiwei W ang: Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P. R. China Email address:zhiwei@bnu.edu.cn Chenlong Yue: School ...

work page internal anchor Pith review Pith/arXiv arXiv