AI 中文总结
本文研究 F_2[T] 上首一多项式正交重数为正的比例，通过建立局部剩余类的互反与分布，证明其渐近为 L d^{-1/2}，并给出精确常数与数值验证。
AI 中文摘要
设 A = F_2[T]。A 中次数 d ≥ 1 的首一多项式 f 的正交重数 m(f) 定义为满足 deg g < d、gcd(g,f) = 1 且 g/f 的所有部分商次数均为 1 的 g ∈ A 的个数。Blackburn (1998) 提出如下问题：当 d 趋于无穷时，次数为 d 且满足 m(f) > 0 的首一多项式的比例 p(d) 如何变化。我们对 f 的每个素因子 P 赋予一个 F_2 中的剩余类 ψ_P(f)，该剩余类由微分 dT/(T(T+1)f) 确定，并证明关于该族的两条无条件事实：一条互反关系，即当 f 与 T(T+1) 互素时，所有 ψ_P(f) 之和为零；以及一条精确的局部分布，即当 P 恰好整除 f 时，ψ_P 在可容许补全中的消失比例为精确的 (|P|-2)/(2(|P|-1))，而当 P^e 恰好整除 f 且 e ≥ 2 时，该比例为精确的一半。假设这些局部条件独立汇集，我们得到 p(d) ~ L d^(-1/2)，其中 L = (9/(4√π)) 乘以 A 中次数至少为 2 的所有首一不可约多项式 P 的乘积 (1 + 1/(2|P|)) (1 - 1/|P|)^(1/2)，其数值等于 1.2116120743...。同一欧拉乘积在第二个参数处求值，得到给定次数的所有 m(f) 之和等于 2^d 的精确恒等式，这固定了其归一化。我们给出 1 ≤ d ≤ 33 时 2^d p(d) 的精确值；对于 27 ≤ d ≤ 33，预测值与这些值在 5×10^(-5) 范围内一致。
英文摘要
Let A = F_2[T]. The orthogonal multiplicity m(f) of a monic f in A of degree d >= 1 is the number of g in A with deg g < d, gcd(g,f) = 1, and all partial quotients of g/f of degree one. Blackburn (1998) asked how the proportion p(d) of monic polynomials of degree d with m(f) > 0 behaves as d tends to infinity. We attach to each prime divisor P of f a residue psi_P(f) in F_2 of the differential dT/(T(T+1)f), and prove two unconditional facts about this family: a reciprocity relation, namely that the psi_P(f) sum to zero whenever f is prime to T(T+1), and an exact local distribution, namely that psi_P vanishes for a proportion exactly (|P|-2)/(2(|P|-1)) of the admissible completions when P exactly divides f, and for exactly half of them when P^e exactly divides f with e >= 2. Granting that these local conditions pool independently, we obtain p(d) ~ L d^(-1/2) with L = (9/(4 sqrt(pi))) times the product over the monic irreducibles P of A of degree at least two of (1 + 1/(2|P|)) (1 - 1/|P|)^(1/2), equal to 1.2116120743... The same Euler product evaluated at a second parameter returns the exact identity that the m(f) of a given degree sum to 2^d, which fixes its normalisation. We give the exact value of 2^d p(d) for 1 <= d <= 33; the predicted values agree with these to within 5 10^(-5) for 27 <= d <= 33.
Comments6 pages. Ancillary files: an exact enumeration engine in C and a self-contained verifier in Python
