pith. sign in

arxiv: 1906.08614 · v1 · pith:JSM4RVKJnew · submitted 2019-06-20 · 🧮 math.AG

Field generators in two variables and birational endomorphisms of mathbb{A}²

Pierrette Cassou-Nogu\`es , Daniel Daigle This is my paper

Pith reviewed 2026-05-25 19:09 UTC · model grok-4.3

classification 🧮 math.AG
keywords field generatorsbirational endomorphismsaffine planealgebraic geometrytwo variablesrational mapspolynomial rings
0
0 comments X

The pith

A survey of field generators in two variables includes considerable new material and is paired with a review of birational endomorphisms of the affine plane.

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

The paper brings together two topics that originated in Abhyankar's seminar in the 1970s. The first and longer part examines field generators in two variables and adds a substantial body of previously unpublished results. The second part surveys birational endomorphisms of the affine plane. Readers would value the work because these objects encode basic information about polynomial rings, rational maps, and the geometry of the plane in two dimensions.

Core claim

The authors compile a survey of field generators in two variables that incorporates a considerable amount of new material, together with a survey of birational endomorphisms of the affine plane A^2; both subjects originated in Abhyankar's seminar.

What carries the argument

Field generators in two variables (elements that generate the rational function field in a controlled way) and birational endomorphisms of A^2 (rational maps from the affine plane to itself that admit an inverse of the same type).

Load-bearing premise

The new material presented on field generators consists of original results not reducible to earlier published work.

What would settle it

A search of the cited literature that locates the claimed new results on field generators already published elsewhere would falsify the assertion of considerable new material.

Figures

Figures reproduced from arXiv: 1906.08614 by Daniel Daigle, Pierrette Cassou-Nogu\`es.

Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The next example gives a new family of bad field generators that shows that neither the number of dicriticals nor their degrees are bounded (and we show in 5.15 that these bad field generators are lean). This family generalizes Jan’s polynomial [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: 2.9. Example. Let ϕ(X) = Xn + cn−1Xn−1 + cn−2Xn−2 + · · · + c0 ∈ k[X] = k [1] where n ≥ 4, c0, . . . , cn−1 ∈ k and c0 6= 0. Denote by ϕ˜(X) = 1 + cn−1X + cn−2X 2 + · · · + c0X n the reciprocal polynomial of ϕ. Let FCND(X, Y ) = 1 ϕ˜(Xn−2Y ) (X + cn−1X n−1Y + · · · + cn−iX n−iY (X n−1Y + ˜ϕ(X n−2Y ))i−1 + · · · + c0Y (X n−1Y + ˜ϕ(X n−2Y ))n−1 ) Thus FCND ∈ k(X, Y ); since X + cn−1X n−1Y + · · · + cn−iX (n−… view at source ↗
Figure 3
Figure 3. Figure 3: We shall now study how field generators behave under birational extensions. 2.11. Definition. Let Φ : X → Y be a morphism of nonsingular algebraic surfaces over k. Assume that Φ is birational, i.e., that there exist nonempty Zariski-open subsets U ⊆ X and V ⊆ Y such that Φ restricts to an isomorphism U → V . By a missing curve of Φ we mean a curve C ⊂ Y such that C ∩ Φ(X) is a finite set of closed points. … view at source ↗
Figure 4
Figure 4. Figure 4: It is easy to check that F is a rational polynomial (hence a field generator) of A. Two fibers of F are reducible namely, let h1(x, y) = x + 2 + 8x 9 y 3 + 8x 3 y + 12x 6 y 2 + 2x 12y 4 + 2x 4 y, [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: It follows that F is a rational polynomial (hence a field generator) of A with ∆(F, A) = [9, 2]. We claim: (21) F is a very bad field generator of A that is lean in A [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: Indeed, we have Γalg(F, A) ⊆ {(v),(w)} by 3.5(b); as deg F(v, 0) = 9 and deg F(0, w) = 3, 3.5(a) gives Γalg(F, A) = ∅. Then F is a very bad field generator of A by 4.5. Lemma 5.12(i) implies that F − t is irreducible for every t ∈ k ∗ and that F = P Q is the prime factorization of F; thus subdegγ (F) = degγ (Q) = 31 (where γ = (v, w)) and δ(F, A) = 11 > 63 31 = degγ (F) subdegγ (F) . So 5.11 implies that F… view at source ↗
read the original abstract

This article is a survey of two subjects: the first part is devoted to field generators in two variables, and the second to birational endomorphisms of the affine plane. Each one of these subjects originated in Abhyankar's seminar in Purdue University in the 1970s. Note that the part on field generators is more than a survey, since it contains a considerable amount of new material.

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.

Referee Report

0 major / 2 minor

Summary. This article surveys two subjects originating in Abhyankar's Purdue seminar: field generators in two variables (containing a considerable amount of new material) and birational endomorphisms of the affine plane.

Significance. If the new material on field generators is original and correct, the survey would serve as a useful reference compiling results on field generators and birational maps of A^2, strengthening the literature in affine algebraic geometry.

minor comments (2)
  1. [Abstract] The abstract states that the field-generators part 'contains a considerable amount of new material' without identifying the specific new results or theorems; the introduction should explicitly demarcate the novel contributions from the survey of prior work.
  2. [Introduction] Notation for field generators and birational endomorphisms should be introduced uniformly at the start of each part to aid readers unfamiliar with the 1970s Purdue context.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of our manuscript, including the accurate summary of its scope as a survey with substantial new material on field generators. We are pleased with the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is explicitly framed as a survey of two subjects (field generators in two variables and birational endomorphisms of A^2), with the first part containing new material. The provided abstract and structure contain no derivations, equations, predictions, fitted parameters, or self-citation chains that could reduce any claim to its inputs by construction. No load-bearing steps of the enumerated kinds are identifiable from the text. The claim of 'considerable amount of new material' is presented as original contribution rather than a derived result, and the work is self-contained against external benchmarks with no internal reductions exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

As this is a survey paper with new material indicated but not detailed, the ledger is empty; no free parameters, axioms, or invented entities are extractable from the abstract.

pith-pipeline@v0.9.0 · 5587 in / 1040 out tokens · 23771 ms · 2026-05-25T19:09:48.739222+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

  1. [1]

    Abhyankar

    Shreeram S. Abhyankar. Dicritical divisors and J acobian problem. Indian J. Pure Appl. Math. , 41:77--97, 2010

    work page 2010

  2. [2]

    Cassou-Nogu \`e s

    P. Cassou-Nogu \`e s. Newton trees at infinity of algebraic curves. In Affine algebraic geometry , volume 54 of CRM Proc. Lecture Notes , pages 1--19. Amer. Math. Soc., Providence, RI, 2011

    work page 2011

  3. [3]

    Cassou-Nogu \`e s

    P. Cassou-Nogu \`e s. Bad field generators. In Affine algebraic geometry , volume 369 of Contemp. Math. , pages 77--83. Amer. Math. Soc., Providence, RI, 2005

    work page 2005

  4. [4]

    P.\ Cassou-Nogu \`e s and D. Daigle. Lean factorizations of polynomial morphisms. In preparation

    work page

  5. [5]

    P.\ Cassou-Nogu \`e s and D. Daigle. Very good and very bad field generators. To appear in Kyoto J.\ of Math

    work page

  6. [6]

    P.\ Cassou-Nogu \`e s and D. Daigle. Compositions of birational endomorphisms of the affine plane. To appear in Pacific J.\ of Math

    work page

  7. [7]

    D. Daigle. Birational endomorphisms of the affine plane. J. Math. Kyoto Univ. , 31(2):329--358, 1991

    work page 1991

  8. [8]

    D. Daigle. Local trees in the theory of affine plane curves. J. Math. Kyoto Univ. , 31(3):593--634, 1991

    work page 1991

  9. [9]

    D. Daigle. Generally rational polynomials in two variables. To appear in Osaka J. of Math

    work page

  10. [10]

    C. J. Jan. On polynomial generators of (X,Y) . PhD thesis, Purdue University, 1974

    work page 1974

  11. [11]

    Miyanishi and T

    M. Miyanishi and T. Sugie. Generically rational polynomials. Osaka J.\ Math. , 17:339--362, 1980

    work page 1980

  12. [12]

    W. D. Neumann and P. Norbury. Rational polynomials of simple type. Pacific J. Math. , 204:177--207, 2002

    work page 2002

  13. [13]

    K.P. Russell. Field generators in two variables. J. Math. Kyoto Univ. , 15:555--571, 1975

    work page 1975

  14. [14]

    K.P. Russell. Good and bad field generators. J. Math. Kyoto Univ. , 17:319--331, 1977

    work page 1977

  15. [15]

    Generically rational polynomials of quasi-simple type

    I.\ Sasao. Generically rational polynomials of quasi-simple type. J. Algebra , 298:58--104, 2006

    work page 2006