One map, three public descriptions

One map, three public descriptions#

Status date: 21 July 2026. This note compares the two local paper drafts and the three public presentations they synthesize. It separates mathematical equivalence from provenance. It is not a priority determination.

Short answer#

The constructions are equivalent as presentations of one algebraic morphism: forgetting a marked simple root of a binary cubic in a special affine slice. They are not literally the same coordinate formula. The displayed identification between the polynomial maps usually denoted by \(F\) and \(G\) is left–right linearly equivalent: independent linear changes are made in the source and the target. It is therefore not, as displayed, a dynamical conjugacy by one common coordinate change. No claim about the possible existence of some other nonlinear conjugacy is needed here.

The public roles are distinct:

  1. Levent Alpöge’s post gives the explicit polynomial formula, determinant, and collision, and credits Fable for working on it.

  2. Andy Jiang’s post gives the projective symmetric-product construction and labels the text as GPT output.

  3. Aaron Lou’s note (@aaron_lou) gives a factorization–resultant derivation and an explicit global affine chart.

  4. Daniel Litt (@littmath) later gave a conceptual pencil-bundle gloss on why the projective source is affine three-space, described it as developed with substantial ChatGPT help, and emphasized that the projective and coordinate posts are the same construction. This is an explanation of the existing construction, not a fourth one.

  5. This repository compares, verifies, and explains these presentations. Its compiler does not claim discovery of the formula or either construction.

What the two retained paper artifacts contain#

The live file src/distinguished_root_map.tex is now the revised canonical synthesis. An earlier working copy at that path was a longer direct-coordinate draft; its useful material was absorbed into the canonical paper, and stale descriptions of that working copy should not be applied to the current file. The older coordinate-free paper is retained unchanged as historical version 1.

Feature

Current canonical synthesis

Historical version 1

Artifact

src/distinguished_root_map.tex and output/pdf/one_map_three_descriptions.pdf

output/pdf/marked_root_binary_cubic.tex and .pdf

Opening

Exact formula, determinant, collision, referee-status note

Intrinsic marked-root theorem and incidence variety

Expository order

Plain-language mechanism, then projective, resultant, and coordinate lenses

Coordinate-free construction first, coordinates later

Projective lens

Jiang–GPT symmetric-product map, with Litt’s pencil/affine-line-bundle explanation

Intrinsic affine normalization of Jiang’s projective model

Resultant lens

Lou’s gauge, kernel proof, and global chart

More detailed invariant resultant development and étaleness proof

Explicit lens

Alpöge–Fable formula plus the exact bridge \(G=R_0FS_0\)

Recovers both coordinate orders after the intrinsic proof

Provenance

Role-specific statement on page one and in the bibliography

All three public sources recorded in a provenance paragraph

Best use

Short public entry point and comparison of the three approaches

Deeper coordinate-free reference

There is no mathematical contradiction between the artifacts and no false core identity was found. The canonical synthesis keeps the historical paper’s invariant proof while adding the direct root-marking intuition, an exact source–target equivalence statement, and a shorter beginner-first path.

The shared morphism#

Think of a cubic as three points of the projective line, counted with multiplicity. Mark one root, then forget which root was marked:

cubic + one marked simple root  ───────────────▶  cubic
                                      forget

A general cubic has three simple roots, so it has three possible markings. The marking is deleted when the marked root becomes repeated. This removes ramification upstairs without removing every corresponding limiting value downstairs. The lost inverse branches therefore escape the affine chart instead of meeting at a finite critical point.

Three languages describe that arrow:

Lens

Source object

Map

What it explains fastest

Explicit algebra

\((x,y,z)\in\mathbf A^3\)

Alpöge’s displayed \(F\)

Exact determinant and three-point collision

Projective marked roots

\((p,\{q,r\})\) away from marked collisions

\((p,\{q,r\})\mapsto\{p,q,r\}\)

Degree three, ramification deletion, special hyperplane

Resultant-normalized factors

\(C=\ell q\), \(\operatorname{Res}(\ell,q)=1\)

\((\ell,q)\mapsto\ell q\)

Gauge fixing, affine chart, clean differential proof

The coordinate-free incidence description used by this repository is a synthesis of the last two lenses, not a fourth discovery claim.

Litt’s explanation supplies the missing geometric digestion of the source. After projection to the marked linear-factor class (equivalently, the marked point), the two deleted loci are lines in the quadratic-factor plane; at the tangency fiber they coincide. Passing to their limiting pencils gives a base obtained from \(\mathbf P^1\times\mathbf P^1\) by deleting a horizontal section and the diagonal, hence an affine plane. Removing each pencil’s base point leaves an affine-line fiber. This explains why an \(\mathbf A^1\)-bundle over \(\mathbf A^2\) appears. It does not by itself replace the final trivialization step: the canonical paper retains the explicit chart and its polynomial inverse as the elementary global proof that the total space is \(\mathbf A^3\).

Exact bridge between the projective and resultant gauges#

In the elementary draft, write

\[ L=vT-uS,\qquad Q=AT^2+BTS+CS^2, \]

and normalize

\[ h=vB-uA=-2,\qquad r=Q(u,v)=2. \]

Set

\[ \ell(T,S)=L(T,-S)=vT+uS, \qquad q(T,S)=\tfrac12 Q(T,-S). \]

Lou’s factor coordinates are then

\[ x=v,\quad \beta=u,\quad \gamma=A/2, \quad \delta=-B/2,\quad \varepsilon=C/2. \]

The two normalizations become

\[ x\delta+\beta\gamma=-h/2=1, \]

and

\[ \operatorname{Res}(\ell,q) =x^2\varepsilon-x\beta\delta+\beta^2\gamma =r/2=1. \]

Thus the apparently different gauge conditions are identical after the displayed sign and scale change.

At affine-chart level, put

\[ S_0(x,y,z)=(x,y,-z), \qquad R_0(a,b,c)=(c,b,a). \]

If \(F\) is the explicit map in Alpöge’s post and \(G\) is the polynomial map in the resultant chart, then exactly

\[ \boxed{G=R_0\circ F\circ S_0.} \]

This is left–right equivalence. It preserves étaleness, geometric degree, fiber cardinalities, and nonproperness. The displayed relation is not a conjugacy \(G=L^{-1}FL\), because its source and target changes are different; no stronger nonconjugacy assertion is being made.

What is intrinsic and what is a choice#

Intrinsic data:

  • the forgetful marked-root morphism;

  • deletion of precisely the repeated marked-root locus;

  • generic degree three and the \(3,1,0\) simple-root fiber counts;

  • nonproperness along the discriminant;

  • the omitted triple-root curve.

Coordinate choices:

  • a basis of the two-dimensional vector space of linear forms;

  • the particular tangent-but-not-osculating hyperplane;

  • signs and scalar normalizations of forms and resultants;

  • the order and scaling of the three target coordinates;

  • the numerical constant appearing as the Jacobian determinant.

Every tangent-but-not-osculating hyperplane to the twisted cubic is equivalent under a projective linear change of the underlying projective line. Those choices change the formula, not the marked-root mechanism.

Actual structure of the canonical paper#

The canonical paper has a page-one certificate followed by six numbered sections and no table of contents:

  1. Provenance and the three viewpoints. It assigns the public roles, translates the minimum terminology, and displays one unifying diagram.

  2. Jiang–GPT: mark a root, then forget it. It develops the symmetric-power map, the ramification deletion, and the tangent-but-not-osculating affine slice.

  3. Lou: fix the factorization gauge by a resultant. It factors \(C=\ell q\), imposes resultant one, proves étaleness by the kernel argument, uses Litt’s pencil-bundle gloss to reveal the affine-plane base and affine-line fibers, and gives the global \(\mathbf A^3\) chart.

  4. Alpöge–Fable: recovering the displayed formula. It gives the exact sign-and-scale bridge, displays \(G=R_0FS_0\), and recovers the determinant and collision.

  5. Fibers, the discriminant, and escape to infinity. It gives the \(3,1,0\) set-theoretic counts, distinguishes the image from the properness set, and explains the escaping sheets.

  6. Scope, history, and reproducibility. It records standard literature, the consequence for all dimensions at least three, the still-separate plane case, real-slice visualization limits, licenses, and reproducible artifacts.

This realized ordering keeps the shortest exact certificate on page one, gives a nonexpert the root-marking metaphor before the invariant proof, and lets experts reach the coordinate bridge without carrying two papers at once.

Visualization warning#

The map is defined on \(\mathbf C^3\), which has six real dimensions. Every three-dimensional plot in the website and animation shows only the real slice \(\mathbf R^3\subset\mathbf C^3\). It illustrates exact real curves and points, but it is not a picture of the whole complex morphism.