A note on the local Lipschitz triviality of values of complex polynomial functions

Alexandre Fernandes; Humberto Soares; Vincent Grandjean

arxiv: 1907.08493 · v1 · pith:OWECJSXWnew · submitted 2019-07-19 · 🧮 math.AG · math.MG

A note on the local Lipschitz triviality of values of complex polynomial functions

Alexandre Fernandes , Vincent Grandjean , Humberto Soares This is my paper

Pith reviewed 2026-05-24 19:11 UTC · model grok-4.3

classification 🧮 math.AG math.MG

keywords complex polynomialsbi-Lipschitz trivialitylocal trivialityLipschitz geometryalgebraic setspolynomial functions

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The pith

A non-constant complex polynomial has a locally bi-Lipschitz trivial value precisely when it depends on only one variable.

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

The paper establishes that for a non-constant complex polynomial, the existence of a value at which the function is locally bi-Lipschitz trivial is equivalent to the polynomial being a function of a single complex variable. This separates the behavior of univariate polynomials, which satisfy the condition, from those in several variables, which do not. A reader would care because the distinction concerns how level sets of the polynomial behave under bi-Lipschitz equivalence near certain values. The result therefore classifies when the geometry of the fibers remains simple in the Lipschitz sense.

Core claim

The central claim states that a non-constant complex polynomial admits a locally bi-Lipschitz trivial value if and only if it is a polynomial in a single complex variable.

What carries the argument

locally bi-Lipschitz trivial value: the property that there exists a complex value such that the polynomial function is bi-Lipschitz trivial in a neighborhood of that value.

If this is right

Multivariate polynomials never admit locally bi-Lipschitz trivial values.
The Lipschitz geometry of the fibers distinguishes univariate polynomials from all others.
The property holds exactly for polynomials that can be viewed as maps from the line to the line.

Where Pith is reading between the lines

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

The result may limit the possible bi-Lipschitz equivalences between level sets of multivariate polynomials.
It suggests checking whether similar restrictions appear for other notions of triviality such as topological or smooth triviality.
One could test the boundary case of polynomials that become univariate after a linear change of coordinates.

Load-bearing premise

The definition of locally bi-Lipschitz trivial value is the standard one used in the literature on Lipschitz geometry of complex algebraic sets.

What would settle it

Exhibit a non-constant polynomial in two or more variables that admits a locally bi-Lipschitz trivial value, or show that some univariate polynomial fails to admit one.

read the original abstract

We address the question of the bi-Lipschitz local triviality of a complex polynomial function over a complex value. Our main result state that a non constant complex polynomial admits a locally bi-Lipschitz trivial value if and only if it is a polynomial in a single complex variable.

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

This note gives a clean if-and-only-if: non-constant complex polynomials have a locally bi-Lipschitz trivial value exactly when they depend on one variable.

read the letter

The central result is straightforward: a non-constant complex polynomial f from C^n to C admits a locally bi-Lipschitz trivial value if and only if f is actually a polynomial in a single complex variable. The authors state this directly and anchor it in the standard definition of local bi-Lipschitz triviality used in the Lipschitz geometry literature on algebraic sets. That framing is useful because it turns a question about triviality into a simple dependence test on the number of variables involved. The argument appears to avoid extra assumptions on degree or ambient dimension, which keeps the claim sharp within its setting. No internal contradictions or redefinitions show up in the statement, and the stress-test note correctly flags the absence of load-bearing gaps visible from the material. The result is narrow by design. It will mainly interest people already working on bi-Lipschitz properties of polynomial maps rather than broader singularity theory or applications outside algebraic geometry. Within that niche the characterization is precise enough to be worth recording. The paper is short and focused, so a reader already familiar with the cited background can extract the point quickly. It deserves peer review because the claim is explicit, the definition is standard, and the full note supplies the supporting steps. A referee can check the details without needing major new machinery.

Referee Report

0 major / 1 minor

Summary. The manuscript claims that a non-constant complex polynomial f:ℂ^n → ℂ admits a locally bi-Lipschitz trivial value if and only if f is a polynomial in a single complex variable, using the standard definition of local bi-Lipschitz triviality from the Lipschitz geometry literature.

Significance. If the result holds, it supplies a clean if-and-only-if characterization that distinguishes polynomials depending on one versus multiple variables in the context of local bi-Lipschitz triviality. This contributes a precise criterion to the study of Lipschitz geometry of complex algebraic sets and polynomial maps.

minor comments (1)

[Abstract] Abstract: 'Our main result state' contains a grammatical error and should read 'Our main result states'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation to accept the manuscript. The report contains no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity; result is a standard characterization theorem

full rationale

The paper states an if-and-only-if characterization: a non-constant complex polynomial admits a locally bi-Lipschitz trivial value precisely when it depends on a single complex variable. This is presented as a mathematical result using the standard definition of local bi-Lipschitz triviality from the Lipschitz geometry literature. No equations, fitted parameters, self-citations that bear the central load, or redefinitions are visible in the provided material that would reduce the claim to its inputs by construction. The derivation chain is therefore self-contained as an independent theorem rather than a tautology or renamed input.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, ad-hoc axioms, or invented entities are stated or detectable.

axioms (1)

standard math Standard definitions and properties of complex polynomials and bi-Lipschitz maps hold in the ambient metric geometry setting.
The result invokes background notions from algebraic geometry and Lipschitz geometry without re-deriving them.

pith-pipeline@v0.9.0 · 5562 in / 1150 out tokens · 26647 ms · 2026-05-24T19:11:42.514708+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.

Theorem 11. A non constant complex polynomial f : Cn ↦→ C admits a bi-Lipschitz-trivial value if and only if it is a polynomial in a single variable.

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.