AI 中文总结
本文研究有限域上多项式环中数字受限的不可约多项式计数，利用傅里叶分析得到渐近公式，并推广圆法至任意多项式基。
AI 中文摘要
我们研究 $\mathbb{F}_q[T]$ 上的首一不可约多项式，其非首项数字（相对于任意多项式基 $b(T)$）避开一个预先指定的禁止数字集合。将数字集合与环 $D=\mathbb{F}_q[T]/(b)$ 等同，我们在允许数字集合的傅里叶参数条件下，得到了此类不可约多项式数量的渐近公式。主项包含一个奇异级数，用于度量允许数字中单位元的相对密度，而误差项则由逐点和平均傅里叶估计共同控制。作为推论，我们得到了一个仅依赖于禁止集合基数的通用判据，以及针对结构化限制的更强结果，包括禁止集合包含所有数字正比例的例子。特别地，我们处理了加法陪集、与 $D$ 的中国剩余分解相容的限制，以及逐系数限制。证明将函数域圆法从受限系数推广到任意多项式基。
英文摘要
We study monic irreducible polynomials over $\mathbb{F}_q[T]$ whose non-leading digits, with respect to an arbitrary polynomial base $b(T)$, avoid a prescribed set of forbidden digits. Identifying the digit set with the ring $D=\mathbb{F}_q[T]/(b)$, we obtain an asymptotic formula for the number of such irreducible polynomials under conditions given in terms of Fourier parameters of the allowed digit set. The main term contains a singular series measuring the relative density of units among the allowed digits, while the error term is controlled by both pointwise and averaged Fourier estimates. As a consequence, we obtain a general criterion depending only on the cardinality of the forbidden set, as well as stronger results for structured restrictions, including examples in which the forbidden set contains a positive proportion of all digits. In particular, we treat additive cosets, restrictions compatible with the Chinese remainder decomposition of $D$, and coefficient-wise restrictions. The proof adapts the function field circle method for restricted coefficients to arbitrary polynomial bases.
Comments31 pages
