二次整数环上的13个未知元与卢卡斯同余式
$13$ unknowns over quadratic integer rings and Lucas congruences
- School of Mathematical Sciences, Peking University（北京大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对二次数域，证明了整数元组的3个未知元丢番图定义，得到13个未知元的递归可枚举整数关系表示，进而得出存在次数界D₀使得无法判定对应多项式方程在二次整数环上是否有解的结论。
AI中文摘要：
对每个二次数域K，我们证明了O_K中整数元组的一致3个未知元丢番图定义，允许有限个多项式非零条件。这给出了有效的+3转移原理，以及每个递归可枚举整数关系的13个未知元表示。因此存在绝对次数界D₀≥1，使得对每个二次数域K，不存在算法，给定P(Y₁,…,Y₁₃)∈ℤ[Y₁,…,Y₁₃]且deg P≤D₀，判定P=0在O_K¹³中是否有解。算术输入是四阶佩尔-卢卡斯同余式，它是范数1的卢卡斯乘法公式的特殊化，该公式给出卢卡斯商与其线性项偏差的精确赋值，以及卢卡斯-维弗里希素数和沃尔-孙-孙素数的偏差准则。我们还建立了范数1的卢卡斯序列二阶修正项的局部满射性与ℓ进密度。
英文摘要：
For every quadratic number field $K$, we prove a uniform $3$-unknown Diophantine definition of integer tuples in $\mathcal{O}_K$, 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 $D_0\geq1$ such that for every quadratic number field $K$, there is no algorithm that, given \[ P(Y_1,\ldots,Y_{13})\in\mathbb{Z}[Y_1,\ldots,Y_{13}],\quad \mathrm{deg}\ P\leq D_0, \] decides whether $P=0$ has a solution in $\mathcal{O}_K^{13}$. 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 $\ell$-adic density for second-order correction terms for norm-one Lucas sequences.
