Family: norm-form — Norm form equation

diophantine_classifier/data/families/norm-form.yaml

@@ -0,0 +1,24 @@
+# Family: Norm 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: norm-form
+name: Norm form equation
+priority: 2
+status: partial
+class: normform
+form: N_{K/QQ}(x_1*w_1 + ... + x_k*w_k) = m
+parents:
+- decomposable-form
+matcher: false
+methods:
+- Schmidt subspace theorem (finiteness)
+- unit-group orbits (full modules)
+software:
+  pari: bnfisintnorm
+  magma: NormEquation
+  sage: K.elements_of_norm(m)
+references:
+- key: Schmidt1972
+  why: finiteness for nondegenerate norm form equations via the subspace theorem
+- key: FinckePohst1985
+  why: the enumeration algorithm behind NormEquation and bnfisintnorm

docs/FAMILIES.md

@@ -281,6 +281,15 @@ conditions, ineffective in general (subspace theorem).
 **References.** Evertse–Győry, *Discriminant Equations in Diophantine Number Theory*
 (2017) and *Unit Equations* (2015).
 
+### `norm-form` — Norm form equations — P2, partial/algorithmic
+**Form.** N_{K/ℚ}(x₁ω₁ + ⋯ + x_kω_k) = m.
+**Status.** Schmidt 1971: finiteness iff nondegenerate (subspace theorem,
+ineffective); full-module case: solutions = finitely many orbits under the unit
+group, computable (this is the algorithmic core of `bnfisintnorm`); effective
+results for special modules (Győry).
+**Software.** PARI: `bnfisintnorm`; Magma: `NormEquation`; Sage: `K.elements_of_norm`.
+**References.** Schmidt 1971; Győry 1980s; Fincke–Pohst 1985.
+
 
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