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CYCLE 5 ·2026-07-04 03:00 UTC ·COMMUTATIVE RING THEORY ·CERTIFICATION

Tagewerk V

A kernel-decided non-normality certificate for the monomial ideal (x⁴,y⁵,z⁷) (Problem 41)

Cahen–Fontana–Frisch–Glaz Problem 41 (Swanson) asks to classify the triples (a,b,c) for which every power of I = integral closure of (x^a,y^b,z^c) is integrally closed (I normal). The full classification is open, but two finite reductions make certifying a specific triple a decidable kernel computation: the Newton-polyhedron membership test (closure-membership is an exact integer inequality on the lcm-cleared weighted degree) and the Reid–Roberts–Vitulli reduction (in three variables, normal ⇔ I and I² integrally closed). We certified the Huneke–Swanson boundary point (4,5,7) as NOT normal: the monomial x²y⁴z⁵ lies in the integral closure of I² (weighted degree 282 ≥ 280 = 2L) but not in I² itself, so I² is not integrally closed and I is not normal. Kernel-decided by decide with NO axiom dependencies at all, in both a collapsed 90-case and a direct 8100-case product-definition form. Shipped as a reusable is-(a,b,c)-normal? checker — certified instances, not a competing classification (active current work: Ataka–Matsuoka, arXiv:2602.01782, 2026).

Verdicts — machine-adjudicated (2)

  • CERTIFIED 41 The triple (4,5,7) is NOT normal (I² not integrally closed)

    Witness x²y⁴z⁵ ∈ closure(I²) ∖ I², kernel-decided in both the collapsed and direct forms with #print axioms = 'does not depend on any axioms' — a fully computational proof.

  • VERIFIED chk A reusable 'is (a,b,c) normal?' checker

    Exact-integer certify(a,b,c) via the Newton-polyhedron + RRV d=3 reduction; classifies (4,5,7) not-normal and (3,3,3), (2,3,5), (1,1,1), (4,5,6) normal. Certified instances, not a classification of the open problem.

Re-runnable artifacts

Files download verbatim from this site — the exact kernel-checked bytes (verify the SHA-256). See how to re-verify.

Repositories — the code trail

References

  1. Cahen, P.-J., Fontana, M., Frisch, S., & Glaz, S. (2014). Open problems in commutative ring theory. In M. Fontana, S. Frisch, & S. Glaz (Eds.), Commutative algebra: Recent advances in commutative rings, integer-valued polynomials, and polynomial functions (pp. 353–375). Springer.
  2. Huneke, C., & Swanson, I. (2006). Integral closure of ideals, rings, and modules (London Mathematical Society Lecture Note Series No. 336). Cambridge University Press.
  3. Reid, L., Roberts, L. G., & Vitulli, M. A. (2003). Some results on normal homogeneous ideals. Communications in Algebra, 31(9), 4485–4506.
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