论文标题
在有限场上随机多项式的低度因素的通用性
Universality for low degree factors of random polynomials over finite fields
论文作者
论文摘要
我们表明，超过$ \ mathbb {f} _q $的低度不可约合因子的计数具有独立但不均匀的系数的行为，就像均匀的随机多项式一样，表现出一种在有限磁场上随机多项式的普遍性的形式。我们最强的结果需要对参数的各种假设，但是我们能够获得仅需要$ q = p $ a Prime的结果，其中$ p \ leq \ exp（{n^{1/13}}）$，其中$ n $是多项式的程度。我们的证明使用傅立叶分析，并依靠Breuillard和Varjú最近应用的工具来研究$ AX+B $过程，该工艺显示了一个点的$ F（α）$。我们将其扩展为处理多个根和$ f $的Hasse衍生物，这使我们能够以多种多样的态度研究不可约合的因素。
We show that the counts of low degree irreducible factors of a random polynomial $f$ over $\mathbb{F}_q$ with independent but non-uniform coefficients behave like that of a uniform random polynomial, exhibiting a form of universality for random polynomials over finite fields. Our strongest results require various assumptions on the parameters, but we are able to obtain results requiring only $q=p$ a prime with $p\leq \exp({n^{1/13}})$ where $n$ is the degree of the polynomial. Our proofs use Fourier analysis, and rely on tools recently applied by Breuillard and Varjú to study the $ax+b$ process, which show equidistribution for $f(α)$ at a single point. We extend this to handle multiple roots and the Hasse derivatives of $f$, which allow us to study the irreducible factors with multiplicity.
