diff --git a/diophantine_classifier/data/families/norm-form.yaml b/diophantine_classifier/data/families/norm-form.yaml new file mode 100644 index 0000000..0e9cdca --- /dev/null +++ b/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 diff --git a/docs/FAMILIES.md b/docs/FAMILIES.md index 86434cf..4c01506 100644 --- a/docs/FAMILIES.md +++ b/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. + ---