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Sums of squares on curves and surfaces

T0 review · 0 major / 3 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read Real algebras R[x,y,sqrt(f1),...,sqrt(fn)] have infinite 2s-Pythagoras numbers under mild assumptions on the fi.

desk verdict The paper shows infiniteness of 2s-Pythagoras numbers for certain real algebras with square roots, finiteness for 0-regulous rings, and codim 2 for the bad set of order 2n. read the letter →

arxiv 2606.22401 v1 pith:J3ZUIGMH submitted 2026-06-21 math.AC math.AG

Bart{l}omiej Bychawski , Bartosz G{l}owacki , Tomasz Kowalczyk This is my paper · ORCID
classification math.ACmath.AG
keywords sumsofsquaresPythagorasnumberrealalgebrasregulousfunctionssingularcurvesevenpowerscodimension
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper first examines sums of higher even powers in the coordinate rings of singular planar curves defined by x^M = y^m with coprime m and M. It then proves that the 2s-Pythagoras number becomes infinite for real algebras obtained by adjoining square roots of polynomials in R[x,y], provided the polynomials satisfy mild conditions. It establishes finiteness of all higher even Pythagoras numbers for the ring of 0-regulous functions on any 0-regulous variety. Finally, it shows that the bad set of order 2n for n greater than 1 can have codimension 2, in contrast to the quadratic sums-of-squares case.

What carries the argument

The 2s-Pythagoras number of an algebra, the smallest integer p such that every sum of 2s-th powers equals a sum of p such powers.

What would settle it

An explicit choice of polynomials f1,...,fn in R[x,y] together with a concrete even integer 2s for which every sum of 2s-th powers in the algebra is a sum of a bounded number of them.

Watch

Extended reading notes

Core claim

We show that the 2s-Pythagoras number of real algebras of the form R[x,y,sqrt(f1),...,sqrt(fn)] are infinite, under some mild assumptions on the polynomials f1,...,fn in R[x,y]. We prove that all of the higher even Pythagoras numbers are finite for the ring of 0-regulous functions on a 0-regulous variety. We then show that the codimension of the bad set of order 2n, for n>1, can be of codimension 2, contrary to the quadratic case.

Load-bearing premise

The mild assumptions on the polynomials f1 through fn in R[x,y] hold and the definitions of 0-regulous functions and varieties match those standard in the literature.

Editorial extensions

If this is right

  • Higher even Pythagoras numbers can be infinite already in real algebras of dimension two.
  • All higher even Pythagoras numbers remain finite on the ring of 0-regulous functions.
  • The locus of non-representable sums of 2n-th powers can drop to codimension two for n>1.
  • Singular curves x^M = y^m provide concrete examples where sums of higher powers behave differently from the smooth case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Adjoining square roots can create new obstructions that prevent bounded-length representations of sums of even powers.
  • The contrast between infinite Pythagoras numbers in the algebraic case and finite ones for regulous functions suggests a sharp distinction between algebraic and regulous positivity.
  • The codimension-two result may indicate that higher-power positivity problems require different geometric tools than the quadratic case.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript studies sums of higher even powers in the coordinate rings of singular planar curves defined by x^M = y^m (coprime positive integers m < M). It proves that the 2s-Pythagoras numbers of the real algebras R[x,y,√f1,…,√fn] are infinite under mild assumptions on the polynomials fi ∈ R[x,y]. It establishes finiteness of all higher even Pythagoras numbers for the ring of 0-regulous functions on a 0-regulous variety. It further shows that the codimension of the bad set of order 2n (n>1) can equal 2, in contrast to the quadratic case.

Significance. If the results hold, the work supplies new infinitude examples for higher-power Pythagoras numbers on singular curves and adjoining square roots, together with a positive finiteness theorem for 0-regulous rings and a codimension phenomenon that diverges from the quadratic setting. The explicit statement of mild assumptions and reliance on standard 0-regulous notions make the claims directly testable against the literature.

minor comments (3)
  1. [Introduction] The introduction should include a brief definition or reference for the 2s-Pythagoras number before its first use in the abstract and §1.
  2. [§1] Notation for the coordinate ring of the curve x^M = y^m is introduced without an explicit equation label; adding (1.1) would aid cross-references in later sections on sums of powers.
  3. [§3] The mild assumptions on f1,…,fn are stated in the abstract and §3 but would benefit from a numbered list or displayed box for quick reference when the infinitude theorem is proved.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and the recommendation for minor revision. The referee's summary correctly reflects the main results on infinitude of 2s-Pythagoras numbers for the indicated real algebras, finiteness for 0-regulous rings, and the codimension-2 phenomenon for bad sets of order 2n.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The abstract and stated claims describe proofs of infinitude for 2s-Pythagoras numbers on specific real algebras under explicit mild assumptions on the fi, finiteness results for 0-regulous rings, and a codimension statement. No equations, definitions, or cited results in the provided text reduce any central claim to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. The 0-regulous notions are flagged as standard in the literature, and the mild assumptions are part of the claim rather than hidden inputs. The derivation chain is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

No information on free parameters, axioms or invented entities is available from the abstract alone.

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Pith. "Pith review of Sums of squares on curves and surfaces." pith.science (2026). https://pith.science/paper/J3ZUIGMH

@misc{pith2026260622401,
  author       = {Pith},
  title        = {Pith review of: Sums of squares on curves and surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J3ZUIGMH}},
  note         = {Machine review of arXiv:2606.22401}
}
abstract

We study sums of higher even powers in the coordinate rings of singular planar curves $x^M=y^m$ for coprime positive integers $m<M$. We then show that the $2s$-Pythagoras number of real algebras of the form $\mathbb{R}[x,y,\sqrt{f_1},\sqrt{f_2},\dots, \sqrt{f_n}]$ are infinite, under some mild assumptions on the polynomials $f_1,f_2,\dots, f_n \in \mathbb{R}[x,y]$. We prove that all of the higher even Pythagoras numbers are finite for the ring of $0$-regulous functions on a $0$-regulous variety. We then show that the codimension of the bad set of order $2n$, for $n>1$, can be of codimension $2$, contrary to the quadratic case.

Discussion (0). Continue with ORCID to comment.

Reference graph

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