Inner and Partial non-degeneracy of mixed functions

Benjamin Bode; Eder L. Sanchez Quiceno

arxiv: 2306.02905 · v1 · submitted 2023-06-05 · 🧮 math.AG · math.CV· math.GT

Inner and Partial non-degeneracy of mixed functions

Benjamin Bode , Eder L. Sanchez Quiceno This is my paper

Pith reviewed 2026-05-24 07:53 UTC · model grok-4.3

classification 🧮 math.AG math.CVmath.GT

keywords mixed polynomialspartial non-degeneracyinner non-degeneracyisolated singularitiesMilnor fibrationsingularity theoryalgebraic geometry

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The pith

Partial non-degeneracy of mixed polynomials implies weakly isolated singularities, and strong inner non-degeneracy produces an explicit Milnor fibration.

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

The paper extends Mondal's partial non-degeneracy from holomorphic polynomials to mixed polynomials, which are polynomial maps from C squared to C that also involve complex conjugates and thus correspond to real maps from R four to R two. It establishes that partial non-degeneracy yields a weakly isolated singularity at the origin while strong partial non-degeneracy yields an isolated singularity. The work further shows that these notions are not equivalent to inner non-degeneracy for mixed functions, in contrast to the holomorphic case, and identifies extra conditions under which equivalence holds. Strongly inner non-degenerate mixed polynomials are proved to satisfy the strong Milnor condition, which supplies an explicit Milnor sphere fibration. Readers care because the results give concrete criteria for detecting isolated singularities and constructing fibrations in this four-dimensional real setting.

Core claim

For mixed polynomials the authors prove that partial non-degeneracy implies a weakly isolated singularity while strong partial non-degeneracy implies an isolated singularity; they also prove that strong inner non-degeneracy implies the strong Milnor condition and therefore an explicit Milnor fibration. Unlike the holomorphic case, partial and inner non-degeneracy are not equivalent for mixed polynomials, though additional conditions restore equivalence between strong partial non-degeneracy and the existence of an isolated singularity.

What carries the argument

The notions of partially non-degenerate, strongly partially non-degenerate, inner non-degenerate, and strongly inner non-degenerate mixed functions, which generalize the corresponding holomorphic concepts and link non-degeneracy to isolation of singularities and to the Milnor condition.

If this is right

Mixed polynomials that meet the partial non-degeneracy condition possess a weakly isolated singularity at the origin.
Strong partial non-degeneracy forces the singularity at the origin to be isolated.
Strong inner non-degeneracy guarantees that the mixed polynomial satisfies the strong Milnor condition and therefore possesses an explicit Milnor sphere fibration.
Under supplementary conditions that the authors identify, strong partial non-degeneracy becomes equivalent to the existence of an isolated singularity.

Where Pith is reading between the lines

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

The non-equivalence between partial and inner non-degeneracy may require separate computational checks when studying real four-dimensional singularities that arise from mixed equations.
The explicit Milnor fibration constructed under strong inner non-degeneracy could be used to compute topological invariants of the link of the singularity directly from the polynomial coefficients.
One could test the additional equivalence conditions on families of mixed polynomials obtained by adding small conjugate terms to holomorphic examples.

Load-bearing premise

The definitions of partial and inner non-degeneracy carry over from the holomorphic setting to mixed polynomials without inconsistencies or hidden extra restrictions on the functions.

What would settle it

A concrete mixed polynomial that satisfies strong partial non-degeneracy yet whose zero set fails to be an isolated singularity at the origin, or one that is strongly inner non-degenerate yet does not admit a Milnor fibration.

Figures

Figures reproduced from arXiv: 2306.02905 by Benjamin Bode, Eder L. Sanchez Quiceno.

**Figure 2.** Figure 2: a) The parametrized loop v(t). b) The graph of ∂ arg(v) ∂t . Since gt has the required even symmetry, we obtain a corresponding inner 16 [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** Figure 3: Parts of an example graph of ∂ arg(vj (t)) dt . (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

**Figure 4.** Figure 4: Possible graphs of (s3,f )i(u(r, t), r, t) as functions of t for a fixed small positive value of r. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗

read the original abstract

Mixed polynomials $f:\mathbb{C}^2\to\mathbb{C}$ are polynomial maps in complex variables $u$ and $v$ as well as their complex conjugates $\bar{u}$ and $\bar{v}$. They are therefore identical to the set of real polynomial maps from $\mathbb{R}^4$ to $\mathbb{R}^2$. We generalize Mondal's notion of partial non-degeneracy from holomorphic polynomials to mixed polynomials, introducing the concepts of partially non-degenerate and strongly partially non-degenerate mixed functions. We prove that partial non-degeneracy implies the existence of a weakly isolated singularity, while strong partial non-degeneracy implies an isolated singularity. We also compare (strong) partial non-degeneracy with other types of non-degeneracy of mixed functions, such as (strong) inner non-degeneracy, and find that, in contrast to the holomorphic setting, the different properties are not equivalent for mixed polynomials. We then introduce additional conditions under which strong partial non-degeneracy becomes equivalent to the existence of an isolated singularity. Furthermore, we prove that mixed polynomials that are strongly inner non-degenerate satisfy the strong Milnor condition, resulting in an explicit Milnor (sphere) fibration.

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 generalizes Mondal's partial non-degeneracy to mixed polynomials, proves some isolation implications, and shows the notions are not equivalent (unlike holomorphic case).

read the letter

The main new material is the extension of partial non-degeneracy and strong partial non-degeneracy to mixed polynomials, together with the proof that these imply weakly isolated or isolated singularities respectively. It also establishes that these conditions are not equivalent to inner non-degeneracy for mixed functions, which is a clear difference from the holomorphic setting, and adds a result that strong inner non-degeneracy gives the strong Milnor condition with an explicit sphere fibration. Additional conditions are given under which strong partial non-degeneracy recovers isolated singularities. These are direct, stated generalizations with explicit comparison statements. The abstract presents the claims cleanly and the non-equivalence point is the part that stands out as genuinely new. The Milnor fibration claim looks like a useful concrete payoff for people who care about fibrations. The central implications appear to follow from the definitions without visible circularity or hidden restrictions on the support. The main limitation is that only the abstract is available here, so the actual definitions, examples showing non-equivalence, and proof details cannot be checked for edge cases or technical gaps. If the full text supplies concrete mixed polynomial examples and verifies the implications without extra assumptions, the results should hold up. This is narrow work aimed at people already working on singularities of mixed or real polynomials. A reader tracking classifications or Milnor-type results in this subfield would find the statements usable. It deserves a serious referee because the claims are new, build on cited prior definitions, and are presented as falsifiable theorems rather than restatements.

Referee Report

0 major / 3 minor

Summary. The paper generalizes Mondal's notions of partial non-degeneracy and strong partial non-degeneracy from holomorphic polynomials to mixed polynomials f:ℂ²→ℂ (equivalently, real polynomial maps ℝ⁴→ℝ²). It proves that partial non-degeneracy implies a weakly isolated singularity while strong partial non-degeneracy implies an isolated singularity. The notions are shown to be inequivalent to (strong) inner non-degeneracy for mixed polynomials (in contrast to the holomorphic case), with additional conditions identified under which strong partial non-degeneracy is equivalent to the existence of an isolated singularity. Finally, strongly inner non-degenerate mixed polynomials are shown to satisfy the strong Milnor condition, yielding an explicit Milnor sphere fibration.

Significance. If the proofs hold, the results provide concrete non-degeneracy criteria and an explicit fibration for singularities of mixed functions, extending tools from holomorphic singularity theory to the real four-dimensional setting. The explicit Milnor fibration under strong inner non-degeneracy is a concrete, usable contribution.

minor comments (3)

The abstract states multiple theorems and comparisons but the manuscript should include explicit section references (e.g., §3 for the implication proofs, §4 for the inequivalence examples) so readers can locate the statements without searching the full text.
Notation for the support of the mixed polynomial and the precise definitions of 'weakly isolated' versus 'isolated' singularity should be restated at the beginning of the main results section for self-contained reading.
The comparison statements with the holomorphic case would benefit from a short table or enumerated list highlighting where equivalence fails for mixed polynomials.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our generalization of partial non-degeneracy to mixed polynomials and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained generalizations

full rationale

The paper generalizes Mondal's partial non-degeneracy from holomorphic polynomials to mixed polynomials, then proves implications (partial non-degeneracy implies weakly isolated singularity; strong partial non-degeneracy implies isolated singularity) and comparisons with inner non-degeneracy using standard techniques in singularity theory. These are direct mathematical statements with no fitted parameters, no self-definitional loops, and no load-bearing self-citations that reduce the central claims to prior author work. The strong Milnor condition result for strongly inner non-degenerate cases follows from the definitions without circular reduction. The derivation chain rests on external prior definitions and explicit proofs rather than renaming or smuggling ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies entirely on standard mathematical background; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (1)

standard math Standard algebraic and topological properties of polynomials, complex conjugation, and singularity theory
Underpins the definition of mixed polynomials and non-degeneracy conditions.

pith-pipeline@v0.9.0 · 5748 in / 1143 out tokens · 43068 ms · 2026-05-24T07:53:32.886167+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/RealityFromDistinction.lean reality_from_one_distinction unclear

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

We generalize Mondal’s notion of partial non-degeneracy … prove that partial non-degeneracy implies … weakly isolated singularity, while strong partial non-degeneracy implies an isolated singularity … strongly inner non-degenerate mixed polynomials satisfy the strong Milnor condition
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

Definition 3.1 … partially Newton non-degenerate (PND) … systems (s1,f)P = (s2,f)P = (s3,f)P = fP = 0 have no common solution in (C*)²

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.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

R. N. Araújo dos Santos, B. Bode, and E. L. Sanchez Quiceno, Links of singularities of inner non-degenerate mixed functions, arXiv:2208.11655 (2022)

work page arXiv 2022
[2]

R. N. Araújo dos Santos and M. Tibăr,Real map germs and higher open book structures, Geom. Dedicata147 (2010), 177–185

work page 2010
[3]

Bode, Constructing links of isolated singularities of polynomialsR4→ R2, J

B. Bode, Constructing links of isolated singularities of polynomialsR4→ R2, J. Knot Theory Ramifications28 (2019), no. 1, 1950009, 21

work page 2019
[4]

Knot Theory Ramifications29 (2020), no

, Real algebraic links inS3 and braid group actions on the set of n-adic integers, J. Knot Theory Ramifications29 (2020), no. 6, 2050039, 44

work page 2020
[5]

, All links are semiholomorphic, arXiv:2211.12329 (2022)

work page arXiv 2022
[6]

, Closures of T-homogeneous braids are real algebraic , arXiv:2211.15394 (2022)

work page arXiv 2022
[7]

B. Bode, M. R. Dennis, D. Foster, and R. P. King, Knotted fields and explicit fibrations for lemniscate knots, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences473 (2017), no. 2202, 20160829

work page 2017
[8]

Chen,Milnor fibration at infinity for mixed polynomials, Cent

Y. Chen,Milnor fibration at infinity for mixed polynomials, Cent. Eur. J. Math. 12 (2014), no. 1, 28–38

work page 2014
[9]

Fukui and E

T. Fukui and E. Yoshinaga,The modified analytic trivialization of family of real analytic functions, Invent. Math.82 (1985), no. 3, 467–477. 34

work page 1985
[10]

H. C. King,Topological type of isolated critical points, Ann. of Math. (2) 107 (1978), no. 2, 385–397

work page 1978
[11]

A. G. Kouchnirenko,Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), no. 1, 1–31

work page 1976
[12]

Milnor,Singular points of complex hypersurfaces, Annals of Mathematics Studies, No

J. Milnor,Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968

work page 1968
[13]

P.Mondal, How many zeroes? Counting solutions of systems of polynomials via toric geometry at infinity, Springer, Cham, 2021

work page 2021
[14]

Oka,Non-degenerate mixed functions, Kodai Math

M. Oka,Non-degenerate mixed functions, Kodai Math. J.33 (2010), no. 1, 1–62

work page 2010
[15]

J.41 (2018), no

, Łojasiewicz exponents of non-degenerate holomorphic and mixed functions, Kodai Math. J.41 (2018), no. 3, 620–651

work page 2018
[16]

Rudolph, Isolated critical points of mappings from R4 to R2 and a natural splitting of the Milnor number of a classical fibered link

L. Rudolph, Isolated critical points of mappings from R4 to R2 and a natural splitting of the Milnor number of a classical fibered link. I. Basic theory; examples, Comment. Math. Helv.62 (1987), no. 4, 630–645

work page 1987
[17]

C. T. C. Wall,Newton polytopes and non-degeneracy, Journal für die reine und angewandte Mathematik (Crelles Journal)509 (1999), 1–19. B. Bode, Instituto de Ciencias Matemáticas (ICMAT), Consejo Superior de Inves- tigaciones Científicas (CSIC), Campus Cantoblanco UAM, C/ Nicolás Cabrera, 13-15, 28049 Madrid, Spain E-mail address : benjamin.bode@icmat.es E....

work page 1999

[1] [1]

R. N. Araújo dos Santos, B. Bode, and E. L. Sanchez Quiceno, Links of singularities of inner non-degenerate mixed functions, arXiv:2208.11655 (2022)

work page arXiv 2022

[2] [2]

R. N. Araújo dos Santos and M. Tibăr,Real map germs and higher open book structures, Geom. Dedicata147 (2010), 177–185

work page 2010

[3] [3]

Bode, Constructing links of isolated singularities of polynomialsR4→ R2, J

B. Bode, Constructing links of isolated singularities of polynomialsR4→ R2, J. Knot Theory Ramifications28 (2019), no. 1, 1950009, 21

work page 2019

[4] [4]

Knot Theory Ramifications29 (2020), no

, Real algebraic links inS3 and braid group actions on the set of n-adic integers, J. Knot Theory Ramifications29 (2020), no. 6, 2050039, 44

work page 2020

[5] [5]

, All links are semiholomorphic, arXiv:2211.12329 (2022)

work page arXiv 2022

[6] [6]

, Closures of T-homogeneous braids are real algebraic , arXiv:2211.15394 (2022)

work page arXiv 2022

[7] [7]

B. Bode, M. R. Dennis, D. Foster, and R. P. King, Knotted fields and explicit fibrations for lemniscate knots, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences473 (2017), no. 2202, 20160829

work page 2017

[8] [8]

Chen,Milnor fibration at infinity for mixed polynomials, Cent

Y. Chen,Milnor fibration at infinity for mixed polynomials, Cent. Eur. J. Math. 12 (2014), no. 1, 28–38

work page 2014

[9] [9]

Fukui and E

T. Fukui and E. Yoshinaga,The modified analytic trivialization of family of real analytic functions, Invent. Math.82 (1985), no. 3, 467–477. 34

work page 1985

[10] [10]

H. C. King,Topological type of isolated critical points, Ann. of Math. (2) 107 (1978), no. 2, 385–397

work page 1978

[11] [11]

A. G. Kouchnirenko,Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), no. 1, 1–31

work page 1976

[12] [12]

Milnor,Singular points of complex hypersurfaces, Annals of Mathematics Studies, No

J. Milnor,Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968

work page 1968

[13] [13]

P.Mondal, How many zeroes? Counting solutions of systems of polynomials via toric geometry at infinity, Springer, Cham, 2021

work page 2021

[14] [14]

Oka,Non-degenerate mixed functions, Kodai Math

M. Oka,Non-degenerate mixed functions, Kodai Math. J.33 (2010), no. 1, 1–62

work page 2010

[15] [15]

J.41 (2018), no

, Łojasiewicz exponents of non-degenerate holomorphic and mixed functions, Kodai Math. J.41 (2018), no. 3, 620–651

work page 2018

[16] [16]

Rudolph, Isolated critical points of mappings from R4 to R2 and a natural splitting of the Milnor number of a classical fibered link

L. Rudolph, Isolated critical points of mappings from R4 to R2 and a natural splitting of the Milnor number of a classical fibered link. I. Basic theory; examples, Comment. Math. Helv.62 (1987), no. 4, 630–645

work page 1987

[17] [17]

C. T. C. Wall,Newton polytopes and non-degeneracy, Journal für die reine und angewandte Mathematik (Crelles Journal)509 (1999), 1–19. B. Bode, Instituto de Ciencias Matemáticas (ICMAT), Consejo Superior de Inves- tigaciones Científicas (CSIC), Campus Cantoblanco UAM, C/ Nicolás Cabrera, 13-15, 28049 Madrid, Spain E-mail address : benjamin.bode@icmat.es E....

work page 1999