Family: skolem — Zeros of linear recurrences (Skolem problem)

diophantine_classifier/data/families/skolem.yaml

@@ -0,0 +1,20 @@
+# Family: Zeros of linear recurrences (Skolem problem)
+# 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: skolem
+name: Zeros of linear recurrences (Skolem problem)
+priority: 2
+status: partial
+class: expdioph
+form: u_n = 0, u a linear recurrence
+parents:
+- exponential-diophantine
+matcher: false
+notes: Decidable for order <= 4; open for order >= 5 - a decidability-boundary exhibit.
+references:
+- key: Skolem1934
+  why: the p-adic method behind the Skolem-Mahler-Lech theorem
+- key: OuaknineWorrell2012
+  why: 'survey: decidability known only up to order 4'
+- key: Bilu2022
+  why: conditional decision procedure for simple recurrences

docs/FAMILIES.md

@@ -257,6 +257,16 @@ perfect power).
 **References.** Schinzel–Tijdeman 1976; Erdős–Selfridge 1975; Shorey–Tijdeman,
 *Exponential Diophantine Equations* (1986) — the standard reference for this whole section.
 
+### `skolem` — Zeros of linear recurrences (Skolem problem) — P2, partial, decidability open
+**Form.** u_n = 0 where u is a linear recurrence sequence (LRS) of order k.
+**Status.** Zero set = finite ∪ arithmetic progressions (Skolem–Mahler–Lech,
+ineffective p-adic proof); decidable for order ≤ 4 (Mignotte–Shorey–Tijdeman,
+Vereshchagin 1985); **open for order ≥ 5** — a marquee decidability-boundary
+exhibit; recent conditional algorithms for simple LRS (Bilu–Luca–Nieuwveld–
+Ouaknine–Purser–Worrell 2022, assuming p-adic Schanuel + Skolem conjecture).
+**References.** Skolem 1934; Ouaknine–Worrell, "Decision problems for linear
+recurrence sequences" (2012 survey); BLNOPW 2022.
+
 ### `exponential-diophantine` — Purely exponential equations — P2, effective (few terms)
 **Form.** c₁·b₁^{n₁} + ⋯ + c_k·b_k^{n_k} = c (fixed bases, unknown exponents);
 e.g. 2ᵃ + 3ᵇ = 5ᶜ, Goormaghtigh-type, Jeśmanowicz conjecture instances.