$13$ unknowns over quadratic integer rings and Lucas congruences
Abstract
For every quadratic number field K, we prove a uniform $3$-unknown Diophantine definition of integer tuples in 𝒪K, allowing finitely many polynomial nonvanishing conditions. This yields an effective +3 transfer principle and a $13$-unknown representation of every recursively enumerable integer relation. Consequently, there exists an absolute degree bound D0≥1 such that for every quadratic number field K, there is no algorithm that, given P(Y1,…,Y13)∈ℤ[Y1,…,Y13], deg P≤ D0, decides whether P=0 has a solution in 𝒪K13. The arithmetic input is a fourth-order Pell--Lucas congruence. It is a specialization of the norm-one Lucas multiplication formula, which yields exact valuations for the deviation of a Lucas quotient from its linear term, together with deviation criteria for Lucas--Wieferich and Wall--Sun--Sun primes. We also establish local surjectivity and ℓ-adic density for second-order correction terms for norm-one Lucas sequences.
Related
- Estimates of Lucas series and expansions in non-integer bases
- On the Krull dimensions of rings of integer-valued polynomials and polynomial rings
- Stacked-bases Properties, maximal ideals and polynomial closures of Rings of Integer-Valued Polynomials
- Integer-valued polynomials over structural matrix rings
- Infinite families of pairs of real quadratic fields whose class numbers are divisible by a given integer
BibTeX
@misc{zhang2026unknowns,
title = {{$13$ unknowns over quadratic integer rings and Lucas congruences}},
author = {Geng-Rui Zhang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30389},
eprint = {2608.30389},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30389},
}
RIS
TY - UNPB
TI - $13$ unknowns over quadratic integer rings and Lucas congruences
AU - Geng-Rui Zhang
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30389
UR - https://arxiv.org/abs/2608.30389
AB - For every quadratic number field K, we prove a uniform $3$-unknown Diophantine definition of integer tuples in 𝒪_K, allowing finitely many polynomial nonvanishing conditions. This yields an effective +3 transfer principle and a $13$-unknown representation of every recursively enumerable integer relation. Consequently, there exists an absolute degree bound D_0≥1 such that for every quadratic number field K, there is no algorithm that, given P(Y_1,…,Y_13)∈ℤ[Y_1,…,Y_13], deg P≤ D_0, decides whether P=0 has a solution in 𝒪_K^13. The arithmetic input is a fourth-order Pell--Lucas congruence. It is a specialization of the norm-one Lucas multiplication formula, which yields exact valuations for the deviation of a Lucas quotient from its linear term, together with deviation criteria for Lucas--Wieferich and Wall--Sun--Sun primes. We also establish local surjectivity and ℓ-adic density for second-order correction terms for norm-one Lucas sequences.
ER -