From 6a8c63704651c53020182de7f126c877441606b2 Mon Sep 17 00:00:00 2001
From: David Roe <roed.math@gmail.com>
Date: Sun, 9 Aug 2026 16:24:44 -0400
Subject: [PATCH] =?UTF-8?q?Family:=20genus-zero-curve=20=E2=80=94=20Genus?=
 =?UTF-8?q?=200=20curve?=
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One family: **Genus 0 curve** (`genus-zero-curve`) — P1, algorithmic.

- Standard form: `C(x, y) = 0 irreducible of genus 0`
- Parent: `general-curve`

Contents: the registry entry, its `docs/FAMILIES.md` section and its rows in the classification corpus.

References cited (the claim each one supports):

- `PoulakisVoskos2000` — practical algorithms for genus-zero equations

Independent of every other family PR: it needs only the registry backbone it is based on.

Part of the series that splits #1 into reviewable pieces:

1. `01-bibliography` — packaging, docs, annotated bibliography
2. `02-parsing` — equation strings to a term model
3. `03-registry` — the YAML family registry (3 seed families)
4. `04-matchers` — shape recognizers
5. `05-classify` — the classification pipeline
6. `06-solvers` — solver framework, two seed solvers, and the CLI
7. `07..09-backbone` — the 23 parent families of the DAG, by depth
8. one PR per remaining family (38 of them, mutually independent)
9. `99-polish` — restore the full doctests and tighten the invariants
---
 .../data/families/genus-zero-curve.yaml       | 22 +++++++++++++++++++
 docs/FAMILIES.md                              |  9 ++++++++
 tests/test_classify.py                        |  1 +
 3 files changed, 32 insertions(+)
 create mode 100644 diophantine_classifier/data/families/genus-zero-curve.yaml

diff --git a/diophantine_classifier/data/families/genus-zero-curve.yaml b/diophantine_classifier/data/families/genus-zero-curve.yaml
new file mode 100644
index 0000000..7fd9157
--- /dev/null
+++ b/diophantine_classifier/data/families/genus-zero-curve.yaml
@@ -0,0 +1,22 @@
+# Family: Genus 0 curve
+# Schema: see data/families/README.md; prose documentation in docs/FAMILIES.md.
+# References are keys into data/references.bib, each with a 'why' annotation.
+slug: genus-zero-curve
+name: Genus 0 curve
+priority: 1
+status: algorithmic
+class: curve
+form: C(x, y) = 0 irreducible of genus 0
+parents:
+- general-curve
+matcher: true
+methods:
+- conic reduction
+- parametrization
+- Pell-type analysis of integral points
+software:
+  sage: Conic; Curve.parametrization
+  magma: Parametrization
+references:
+- key: PoulakisVoskos2000
+  why: practical algorithms for genus-zero equations
diff --git a/docs/FAMILIES.md b/docs/FAMILIES.md
index 86434cf..70eba33 100644
--- a/docs/FAMILIES.md
+++ b/docs/FAMILIES.md
@@ -210,6 +210,15 @@ PARI `hyperellratpoints`, Magma `Chabauty`, `PointSearch`.
 **References.** Siegel 1929; Faltings 1983; Bombieri–Gubler, *Heights in Diophantine
 Geometry*; Stoll, "Rational points on curves" (survey, 2011).
 
+### `genus-zero-curve` — Genus 0 curves — P1, algorithmic
+**Form.** C(x, y) = 0 irreducible of genus 0.
+**Status.** Rational points: none or a ℙ¹-parametrization (conic step:
+Hasse principle + Cremona–Rusin/Simon); integral points on the affine model:
+reduce along the parametrization to Pell-like/divisor conditions (finite iff ≥ 3
+points at infinity, Siegel; effective — Alvanos–Poulakis give complete algorithms).
+**Software.** Sage: `Conic`, `parametrization`; Magma: `Conic`, `Parametrization`.
+**References.** Hilbert–Hurwitz 1890; Poulakis 2002; Alvanos–Poulakis 2011.
+
 ### `genus-one-curve` — Genus 1 curves (non-Weierstrass models) — P1, algorithmic*
 **Form.** C(x, y) = 0 irreducible of genus 1 (any plane model).
 **Status.** With a rational point: birational to an elliptic curve (Nagell/Riemann–Roch
diff --git a/tests/test_classify.py b/tests/test_classify.py
index e3d6386..565ca1a 100644
--- a/tests/test_classify.py
+++ b/tests/test_classify.py
@@ -31,6 +31,7 @@
     ("y^3 = x^4 + 2", "", "superelliptic"),
     ("x^3*y + y^3*z + z^3*x = 0", "", "general-curve"),     # Klein quartic
     ("x^2*y^2 = x^3 + 1", "", "genus-one-curve"),
+    ("y^2 = x^3", "", "genus-zero-curve"),                  # cuspidal
     # Fermat-type
     ("x^2 + y^4 = z^3", "", "generalized-fermat"),
     ("2*x^3 + 3*y^3 = 5*z^3", "", "generalized-fermat"),