AI 中文总结
该研究针对有限域上的独立Haar随机分数Laurent级数，推导了同时逼近时最小公共分母轮廓的精确更新律，给出了残差分布、跳跃规律等结果，建立了r≥2时的同时公共分母律。
AI 中文摘要
设α₁,…,αᵣ是有限域𝔽_q上独立的Haar随机分数Laurent级数，Lᵣ(n)是能同时消去前n个负系数的多项式分母的最小系数长度。我们证明最小核是一条直线，且在停止时间Tₙ=n+Lᵣ(n)-1后立即显现的残差向量是𝔽_q^r上的独立均匀分布。因此，Lᵣ(n)的跳跃指示符是参数为1−q⁻ʳ的独立伯努利变量；在跳跃的条件下，残差方向是𝔽_q上的ℙʳ⁻¹均匀分布。我们还给出了正跳跃大小的精确核增长时钟，以及深度一致的几何尾界；对于两个级数，跳跃在第一个或第二个核增长时期决定，概率分别为q⁻¹和1−q⁻¹。标记更新律在深度变量中产生精确的二项式和波动律，在系数长度变量中新达到的最小分母长度的密度为(1−q⁻ʳ)/r。对于r=1，这是深度坐标中的经典独立部分商次数律，我们给出了其精确字典。新的概率内容是r≥2时的同时公共分母律。我们还建立了与联合线性复杂度的精确轮廓对应和记录对偶公式。
英文摘要
Let $α_1,\ldots,α_r$ be independent Haar-random fractional Laurent series over $\mathbb{F}_q$, and let $L_r(n)$ be the least coefficient length of a polynomial denominator that simultaneously cancels the first $n$ negative coefficients. We prove that the minimal kernel is a line and that the residual vectors revealed immediately after the stopping times $T_n=n+L_r(n)-1$ are iid uniform on $\mathbb{F}_q^r$. Hence the jump indicators of $L_r(n)$ are iid Bernoulli variables with parameter $1-q^{-r}$; conditionally on a jump, the residual direction is uniform on $\mathbb P^{r-1}(\mathbb{F}_q)$. We also give an exact kernel-growth clock for positive jump sizes and a geometric tail bound uniform in the depth; for two series the jump is decided at the first or second kernel-growth epoch with probabilities $q^{-1}$ and $1-q^{-1}$. The marked renewal law yields exact binomial and fluctuation laws in the depth variable and the density of newly attained minimal denominator lengths $\frac{1-q^{-r}}{r}$ in the coefficient-length variable. For $r=1$ this is the classical iid partial-quotient degree law in the depth coordinate, for which we give an exact dictionary. The new probabilistic content is the simultaneous common-denominator law for $r\ge2$. We also establish exact profile-correspondence and record-duality formulas with joint linear complexity.
Comments22 pages, 3 figures
