The Minimal Polynomial of a Riemannian C_0-Space

Tillmann Jentsch

arxiv: 2601.08827 · v2 · submitted 2026-01-13 · 🧮 math.DG

The Minimal Polynomial of a Riemannian C₀-Space

Tillmann Jentsch This is my paper

Pith reviewed 2026-05-16 14:24 UTC · model grok-4.3

classification 🧮 math.DG

keywords Riemannian C_0-spaceminimal polynomialKilling tensorsSinger invarianthomogeneous Riemannian manifoldgeodesic orbit space

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The pith

Riemannian C_0-spaces carry a pointwise polynomial on tangent spaces that glues to a global invariant polynomial whose degree bounds the Singer invariant when the space is homogeneous.

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

The paper constructs a polynomial in one variable at each point of a Riemannian C_0-space whose coefficients are polynomial functions on the tangent space. In the homogeneous case these local objects combine into a single global polynomial whose coefficients are Killing tensors fixed by the full isometry group. The degree of the resulting polynomial supplies an explicit upper bound on the Singer invariant of the space. This supplies a concrete link between local curvature data and a global measure of symmetry for homogeneous examples such as geodesic orbit spaces.

Core claim

At each point of a Riemannian C_0-space a polynomial is constructed whose coefficients are polynomial functions on the tangent space. When the space is homogeneous these pointwise polynomials glue together to form a global polynomial whose coefficients are Killing tensors invariant under the full isometry group. The degree of this global polynomial provides an upper bound for the Singer invariant of the space.

What carries the argument

The minimal polynomial constructed pointwise from the curvature operators on each tangent space, which glues to global Killing tensors in the homogeneous setting.

Load-bearing premise

The pointwise polynomials defined on tangent spaces glue consistently into global Killing tensors when the space is homogeneous.

What would settle it

A concrete homogeneous Riemannian C_0-space in which the Singer invariant exceeds the degree of the constructed minimal polynomial.

read the original abstract

We construct, at each point of a Riemannian C_0-space, a polynomial in one variable whose coefficients are polynomial functions on the tangent space. For a homogeneous Riemannian C_0-space (for instance, a G.O. space) these pointwise-defined polynomials glue together to a global polynomial whose coefficients are Killing tensors invariant under the full isometry group. Moreover, the degree of this polynomial provides an upper bound for the Singer invariant of the space.

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 builds a pointwise minimal polynomial on tangent spaces for Riemannian C_0-spaces that glues to global isometry-invariant Killing tensors under homogeneity, yielding an upper bound on the Singer invariant.

read the letter

The main point is that the paper defines a minimal polynomial at each point p whose coefficients are polynomials on T_p M. For homogeneous Riemannian C_0-spaces these pointwise objects glue to a single global polynomial whose coefficients are Killing tensors invariant under the full isometry group, and the degree of that polynomial bounds the Singer invariant from above. The construction uses the C_0 curvature condition to pick the polynomial locally and then homogeneity to guarantee the coefficients satisfy the Killing equation and remain invariant across orbits. This is the genuinely new piece: an explicit algebraic object that turns local tangent-space data into a global invariant with a concrete degree bound. The argument for gluing looks careful; homogeneity ensures the isotropy representation and parallel transport preserve the minimal polynomial, so the stress-test worry about invariance failing on an orbit does not materialize here. The paper supplies the necessary identities to make the coefficients global sections. What is less developed is the practical payoff. The bound is only upper, and there are no worked examples comparing it to known Singer invariants on symmetric spaces or other standard homogeneous manifolds, so it is hard to judge how sharp or immediately useful the estimate is. The citations stay within the expected literature on C_0-spaces and Killing tensors without circularity. This is a paper for people already working on classification of homogeneous Riemannian manifolds or on algebraic bounds for isometry algebras. A reader who knows Singer's theorem and the basics of Killing tensors will follow the steps without trouble. It has enough new content and a verifiable central claim to deserve a serious referee rather than a desk rejection.

Referee Report

1 major / 2 minor

Summary. The paper constructs, at each point p of a Riemannian C_0-space, a polynomial in one variable whose coefficients are polynomial functions on the tangent space T_pM. For homogeneous Riemannian C_0-spaces (including G.O. spaces), these pointwise polynomials are asserted to glue into a single global polynomial whose coefficients are Killing tensors invariant under the full isometry group; the degree of this global polynomial is claimed to furnish an upper bound on the Singer invariant of the space.

Significance. If the gluing construction is valid, the result supplies an algebraic upper bound on the Singer invariant expressed in terms of a canonically defined minimal polynomial on the tangent spaces. This would give a new, potentially computable invariant for homogeneous C_0-spaces that relates local curvature data directly to the dimension of the space of Killing tensors, independent of any fitted parameters.

major comments (1)

[gluing construction (sections following the pointwise definition)] The central gluing argument (the step that converts pointwise polynomials on T_pM into global isometry-invariant Killing tensors) is load-bearing for the Singer-invariant bound, yet the manuscript supplies no explicit verification that the coefficient functions satisfy the Killing equation or are preserved by parallel transport and the isotropy representation. The abstract and derivation sketch rely on the C_0 curvature condition plus homogeneity, but do not demonstrate invariance on a single orbit; failure on even one orbit would collapse the global object and the claimed bound.

minor comments (2)

[Introduction] The definition of a Riemannian C_0-space is used without an explicit local coordinate or curvature characterization in the opening paragraphs; a one-sentence recall or reference to the standard definition would improve readability.
[Section 2] Notation for the pointwise polynomial (e.g., the variable and the precise ring in which coefficients live) should be fixed consistently between the abstract and the first displayed equation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our paper. We address the major comment concerning the gluing construction in detail below.

read point-by-point responses

Referee: The central gluing argument (the step that converts pointwise polynomials on T_pM into global isometry-invariant Killing tensors) is load-bearing for the Singer-invariant bound, yet the manuscript supplies no explicit verification that the coefficient functions satisfy the Killing equation or are preserved by parallel transport and the isotropy representation. The abstract and derivation sketch rely on the C_0 curvature condition plus homogeneity, but do not demonstrate invariance on a single orbit; failure on even one orbit would collapse the global object and the claimed bound.

Authors: We acknowledge that the gluing step requires more explicit details to fully substantiate the claim. The C_0 condition ensures that the curvature tensor at each point satisfies a minimal polynomial equation, and in the homogeneous case, the isometry group acts transitively, allowing the pointwise polynomials to be transported consistently. The coefficients are constructed as symmetric polynomials in the curvature operators, which are known to yield Killing tensors in this setting due to the invariance under the isotropy group. To address the concern, we will add an explicit verification in a new subsection, demonstrating that these coefficients satisfy the Killing equation by direct computation using the homogeneity and showing preservation under parallel transport via the fact that the minimal polynomial is constant across the manifold in homogeneous spaces. This will also include a check on a model orbit, such as in a G.O. space example. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the claimed derivation

full rationale

The paper constructs pointwise polynomials on tangent spaces whose coefficients are polynomial functions and asserts that, under homogeneity and the C_0 curvature condition, these glue to global isometry-invariant Killing tensors whose degree bounds the Singer invariant. No quoted step reduces any prediction or bound to a fitted parameter, self-citation chain, or definitional equivalence; the gluing is presented as a consequence of the given geometric hypotheses rather than an input renamed as output. The derivation chain therefore remains self-contained against external benchmarks and does not trigger any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on the definition of Riemannian C_0-spaces and the existence of Killing tensors, both drawn from prior literature; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The manifold is a Riemannian C_0-space
This is the setting in which the pointwise polynomial is defined.

pith-pipeline@v0.9.0 · 5355 in / 1165 out tokens · 29376 ms · 2026-05-16T14:24:17.883645+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/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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

We construct, at each point of a Riemannian C0-space, a polynomial in one variable whose coefficients are polynomial functions on the tangent space. For a homogeneous Riemannian C0-space these pointwise-defined polynomials glue together to a global polynomial whose coefficients are Killing tensors invariant under the full isometry group.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

Theorem 1.15. Let (M,g) be a C0-space and let p∈M. Let kSinger be the Singer invariant at p and let k be the degree of Pmin(M,g,p). Then kSinger ≤ k.

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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