Family: index-form — Index form equation

diophantine_classifier/data/families/index-form.yaml

@@ -0,0 +1,19 @@
+# Family: Index form equation
+# 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: index-form
+name: Index form equation
+priority: 3
+status: partial
+class: normform
+form: I(x_2,...,x_k) = m
+parents:
+- decomposable-form
+matcher: false
+methods:
+- Gaal's algorithms (degrees 3-8)
+references:
+- key: Gyory1976
+  why: effective finiteness for index form equations
+- key: Gaal2019
+  why: algorithms by degree, with implementations

docs/FAMILIES.md

@@ -281,6 +281,13 @@ conditions, ineffective in general (subspace theorem).
 **References.** Evertse–Győry, *Discriminant Equations in Diophantine Number Theory*
 (2017) and *Unit Equations* (2015).
 
+### `index-form` — Index form / power integral bases — P3, algorithmic (low degree)
+**Form.** I(x₂,…,x_k) = ±m (index form of an order in a number field K).
+**Status.** Finite for m fixed (Győry, effective); complete practical algorithms
+for degrees 3–5 and many sextic/octic families (Gaál and school).
+**Software.** Gaál's Magma/Maple codes; case-by-case.
+**References.** Győry 1976; Gaál, *Diophantine Equations and Power Integral Bases*
+(2nd ed. 2019).
 
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