# Reading roadmap: An explicit noncoordinate polynomial\\with affine three-space zero fibre

The manuscript defines a polynomial from a cusp relation and a shifted auxiliary algebra. A local lifting lemma identifies its zero-fibre quotient with a polynomial extension of a two-variable ring. An ambient polynomial identity supplies a critical point on another fibre, and a Jacobian argument is used to rule out a coordinate. Adjoining variables gives the higher-dimensional statements; the appendix records both coordinate maps.

Source SHA-256: `60993c0b524291b7f707e42eec89ec7e910c22760d81f1d3af007c229228c13c`. Indexed 18,775 characters.

Edges point from prerequisite to consumer. Solid proof edges are attributed inferences, not formal certificates; references remain candidates.

## Define the construction and scope

Introduce the cusp polynomials and the coordinate/zero-fibre distinction.

Claims: thm:main, cor:dimension

## Establish the lifting parameter

Replace the cusp presentation by a single free polynomial parameter over the base ring.

Claims: lem:lift

## Identify the fibre and apply the obstruction

Combine the base-algebra shift with lifting and the ambient critical-point identity.

Claims: thm:main, cor:dimension

## Record explicit coordinate maps

Give the forward map, inverse map and Bezout certificate.

Claims: app:coordinates

## Internal claim index

### Theorem 1.1 — `thm:main`

Source lines 103–112.

Use a shift to identify the auxiliary base algebra with C[X,Y], check its unit-ideal condition, and apply the lifting lemma to identify R/(F). Differentiate the ambient identity at the displayed point and use Jacobian invertibility to infer that F is not a coordinate.

- `lem:lift` (proof_dependency; source_audit_annotation): The theorem constructs the lemma’s algebra and applies its polynomial-extension conclusion.
- `eq:Bplane` (proof_dependency; source_audit_annotation): The displayed fibre isomorphism substitutes the identified two-variable base algebra.

Candidate references: `eq:xyz` (internal_reference); `eq:F` (internal_reference); `eq:xyz` (candidate_proof_dependency); `eq:shift` (candidate_proof_dependency); `lem:lift` (candidate_proof_dependency); `eq:cusp` (candidate_proof_dependency); `eq:B` (candidate_proof_dependency); `eq:Bplane` (candidate_proof_dependency); `eq:F` (candidate_proof_dependency); `eq:shiftidentity` (candidate_proof_dependency); `eq:ambient` (candidate_proof_dependency); `external:SY` (external_reference).

### Corollary 1.2 — `cor:dimension`

Source lines 116–121.

Adjoin independent variables to the original quotient presentation. The original critical point persists because all new derivatives vanish, so the same Jacobian obstruction applies in larger dimension.

- `thm:main` (proof_dependency; source_audit_annotation): The corollary preserves the original quotient and critical point after adjoining variables.
- `eq:fiber` (proof_dependency; source_audit_annotation): The quotient presentation is explicitly reused in the higher-dimensional argument.

Candidate references: `eq:fiber` (candidate_proof_dependency); `thm:main` (candidate_proof_dependency).

### Lemma 2.1 — `lem:lift`

Source lines 164–176.

Choose a Bezout identity for h and y, substitute G=V+UW, eliminate W using the compatible equations with coefficients h and y², and write inverse B-algebra maps for the remaining linear relation.


Candidate references: `eq:Clinear` (candidate_proof_dependency); `eq:compat` (candidate_proof_dependency); `eq:bezout` (candidate_proof_dependency); `eq:UG` (candidate_proof_dependency); `eq:Wlinear` (candidate_proof_dependency).

## External inputs and references

### [1] AM — `external:AM`

Source lines 383–389.

S.~S.~Abhyankar and T.~T.~Moh, Embeddings of the line in the plane, J.~Reine Angew. Math. 276 (1975), 148--166. \href{https://doi.org/10.1515/crll.1975.276.148}{doi:10.1515/crll.1975.276.148}.

Earlier affirmative coordinate theorem in ambient dimension two.

Role: **background**. Expanded native manuscript lines 47–55. Source role only; the cited original work has not been independently checked for this neutral guide.

- `sec:introduction` (background; source_audit_annotation): Earlier affirmative coordinate theorem in ambient dimension two.

Candidate references: `sec:introduction` (external_reference).

### [2] EEO — `external:EEO`

Source lines 389–395.

M.~El~Kahoui, N.~Essamaoui, and M.~Ouali, Interpolation in the automorphism group of a polynomial ring, Algebra Colloq. 27 (2020), no.~3, 587--598. \href{https://doi.org/10.1142/S1005386720000486}{doi:10.1142/S1005386720000486}.

Further algebraic treatments of restricted affine-space-fibre equations.

Role: **background**. Expanded native manuscript lines 67–68. Source role only; the cited original work has not been independently checked for this neutral guide.

- `sec:introduction` (background; source_audit_annotation): Further algebraic treatments of restricted affine-space-fibre equations.

Candidate references: `sec:introduction` (external_reference).

### [3] Ghosh — `external:Ghosh`

Source lines 395–400.

P.~Ghosh, On $\Aff^1$-contractibility of certain simple birational extensions of affine spaces, \href{https://arxiv.org/abs/2506.09166v4}{arXiv:2506.09166v4} (2026).

An affirmative characteristic-zero embedding theorem under specified restrictions on the equation.

Role: **background**. Expanded native manuscript lines 72–79. Source role only; the cited original work has not been independently checked for this neutral guide.

- `sec:introduction` (background; source_audit_annotation): An affirmative characteristic-zero embedding theorem under specified restrictions on the equation.

Candidate references: `sec:introduction` (external_reference).

### [4] GGP — `external:GGP`

Source lines 400–405.

P.~Ghosh, N.~Gupta, and A.~Pal, On embedding of linear hypersurfaces, \href{https://arxiv.org/abs/2405.07205v3}{arXiv:2405.07205v3} (2024).

Linear hypersurfaces with multivariable coefficients subject to additional divisor hypotheses.

Role: **background**. Expanded native manuscript lines 69–71. Source role only; the cited original work has not been independently checked for this neutral guide.

- `sec:introduction` (background; source_audit_annotation): Linear hypersurfaces with multivariable coefficients subject to additional divisor hypotheses.

Candidate references: `sec:introduction` (external_reference).

### [5] GGPthreefold — `external:GGPthreefold`

Source lines 405–412.

P.~Ghosh, N.~Gupta, and A.~Pal, On the family of affine threefolds $a(x)y=F(x,z,t)$, J.~Algebra 700 (2026), 267--288. \href{https://doi.org/10.1016/j.jalgebra.2026.04.008}{doi:10.1016/j.jalgebra.2026.04.008}; \href{https://arxiv.org/abs/2403.17397v2}{arXiv:2403.17397v2}.

Further algebraic treatments of restricted affine-space-fibre equations.

Role: **background**. Expanded native manuscript lines 67–68. Source role only; the cited original work has not been independently checked for this neutral guide.

- `sec:introduction` (background; source_audit_annotation): Further algebraic treatments of restricted affine-space-fibre equations.

Candidate references: `sec:introduction` (external_reference).

### [6] KVZ — `external:KVZ`

Source lines 412–419.

S.~Kaliman, S.~V\'en\'ereau, and M.~Zaidenberg, \emph{Simple birational extensions of the polynomial algebra $\CC^{[3]}$}, Trans. Amer. Math. Soc. 356 (2004), 509--555. \href{https://doi.org/10.1090/S0002-9947-03-03398-1}{doi:10.1090/S0002-9947-03-03398-1}; \href{https://arxiv.org/abs/math/0104204v2}{arXiv:math/0104204v2}.

Earlier affirmative result for a four-variable linear equation with a one-variable coefficient.

Role: **background**. Expanded native manuscript lines 62–66. Source role only; the cited original work has not been independently checked for this neutral guide.

- `sec:introduction` (background; source_audit_annotation): Earlier affirmative result for a four-variable linear equation with a one-variable coefficient.

Candidate references: `sec:introduction` (external_reference).

### [7] Kraft — `external:Kraft`

Source lines 419–425.

H.~Kraft, \emph{\href{https://numdam.org/item/SB_1994-1995__37__295_0/}{Challenging problems on affine $n$-space}}, S\'eminaire Bourbaki, Vol.~1994/95, Expos\'e~802, Ast\'erisque 237 (1996), 295--317.

Embedding-problem survey and relationship with algebraic group actions.

Role: **background**. Expanded native manuscript lines 56–57. Source role only; the cited original work has not been independently checked for this neutral guide.

- `sec:introduction` (background; source_audit_annotation): Embedding-problem survey and relationship with algebraic group actions.

Candidate references: `sec:introduction` (external_reference).

### [8] Maubach — `external:Maubach`

Source lines 425–431.

S.~Maubach, The commuting derivations conjecture, J.~Pure Appl. Algebra 179 (2003), 159--168. \href{https://doi.org/10.1016/S0022-4049(02)00198-6}{doi:10.1016/S0022-4049(02)00198-6}.

Earlier straightening results for the displayed restricted equations.

Role: **background**. Expanded native manuscript lines 62–66. Source role only; the cited original work has not been independently checked for this neutral guide.

- `sec:introduction` (background; source_audit_annotation): Earlier straightening results for the displayed restricted equations.

Candidate references: `sec:introduction` (external_reference).

### [9] MaubachMonomial — `external:MaubachMonomial`

Source lines 431–438.

S.~Maubach, Triangular monomial derivations on $k[X_1,X_2,X_3,X_4]$ have kernel generated by at most four elements, J.~Pure Appl. Algebra 153 (2000), no.~2, 165--170. \href{https://doi.org/10.1016/S0022-4049(99)00084-5}{doi:10.1016/S0022-4049(99)00084-5}.

Triangular invariant-kernel origin of the chosen polynomials; the written proof proceeds directly through the cusp lifting lemma.

Role: **background**. Expanded native manuscript lines 140–145. Source role only; the cited original work has not been independently checked for this neutral guide.

- `sec:introduction` (background; source_audit_annotation): Triangular invariant-kernel origin of the chosen polynomials; the written proof proceeds directly through the cusp lifting lemma.

Candidate references: `sec:introduction` (external_reference).

### [10] Russell — `external:Russell`

Source lines 438–444.

P.~Russell, Simple birational extensions of two dimensional affine rational domains, Compositio Math. 33 (1976), no.~2, 197--208. \href{https://www.numdam.org/item/CM_1976__33_2_197_0/}{Archival full text}.

Earlier extension of a restricted linear-variable embedding result to arbitrary fields.

Role: **background**. Expanded native manuscript lines 58–60. Source role only; the cited original work has not been independently checked for this neutral guide.

- `sec:introduction` (background; source_audit_annotation): Earlier extension of a restricted linear-variable embedding result to arbitrary fields.

Candidate references: `sec:introduction` (external_reference).

### [11] Sathaye — `external:Sathaye`

Source lines 444–450.

A.~Sathaye, On linear planes, Proc. Amer. Math. Soc. 56 (1976), 1--7. \href{https://doi.org/10.1090/S0002-9939-1976-0409472-6}{doi:10.1090/S0002-9939-1976-0409472-6}.

Distinction between the coordinate and all-fibres questions, and earlier affirmative linear-variable cases.

Role: **background**. Expanded native manuscript lines 50–60. Source role only; the cited original work has not been independently checked for this neutral guide.

- `sec:introduction` (background; source_audit_annotation): Distinction between the coordinate and all-fibres questions, and earlier affirmative linear-variable cases.

Candidate references: `sec:introduction` (external_reference).

### [12] SY — `external:SY`

Source lines 450–457.

V.~Shpilrain and J.-T.~Yu, Embeddings of hypersurfaces in affine spaces, J.~Algebra 239 (2001), no.~1, 161--173. \href{https://doi.org/10.1006/jabr.2000.8677}{doi:10.1006/jabr.2000.8677}; \href{https://arxiv.org/abs/math/0010211v1}{arXiv:math/0010211v1}.

Critical-locus obstruction and an earlier question motivating the construction; the Jacobian argument is explained locally.

Role: **background**. Expanded native manuscript lines 122–128. Source role only; the cited original work has not been independently checked for this neutral guide.

- `sec:introduction` (background; source_audit_annotation): Critical-locus obstruction and an earlier question motivating the construction; the Jacobian argument is explained locally.

Candidate references: `sec:introduction` (external_reference); `thm:main` (external_reference).

### [13] Suzuki — `external:Suzuki`

Source lines 457–464.

M.~Suzuki, Propri\'et\'es topologiques des polyn\^omes de deux variables complexes, et automorphismes alg\'ebriques de l'espace $\CC^2$, J.~Math. Soc. Japan 26 (1974), 241--257. \href{https://doi.org/10.2969/jmsj/02620241}{doi:10.2969/jmsj/02620241}.

Earlier affirmative coordinate theorem in ambient dimension two.

Role: **background**. Expanded native manuscript lines 47–55. Source role only; the cited original work has not been independently checked for this neutral guide.

- `sec:introduction` (background; source_audit_annotation): Earlier affirmative coordinate theorem in ambient dimension two.

Candidate references: `sec:introduction` (external_reference).

## Coverage and limits

{"mode": "native_latex_index", "claims": 3, "sections": 4, "external_references": 13, "source_bibliography_entries": 13, "unresolved_bibliography": 0, "unresolved": 0}

- Reference edges are not certified proof dependencies. Edge direction is prerequisite → consumer.
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