论文标题
随机多项式的不可约性：一般度量
Irreducibility of random polynomials: general measures
论文作者
论文摘要
令$μ$为$ \ mathbb {z} $的概率度量，这不是狄拉克质量，并且具有有限的支持。我们证明，如果一元多项式$ f（x）\ in \ mathbb {z} [x] [x] $ $ n $的$由$μ$独立选择，同时确保$ f（0）\ neq0 $，那么$θ= trivis $ f divis $ f（x）$ f（x）$ f（x）$ f（x）$ f（x）。倾向于1 $ n \ to \ infty $。 此外，在某些情况下，我们表明，带有$ f（0）\ neq0 $的随机多项式$ f（x）$是不可约的，概率趋于1 $ n \ to \ to \ hyfty $。特别是，如果$μ$是连续35个连续整数的统一度量，或者是$ [ - h，h] \ cap \ cap \ mathbb {z} $ of Cardinality $ \ ge H^{4/5}（\ log h）（\ log h）^2 $与$ H $足够大。此外，在所有这些设置中，我们表明$ f（x）$的Galois组为$ \ Mathcal {a} _n $或$ \ Mathcal {s} _n $，具有很高的可能性。 最后，当$μ$是至少两个元素的有限算术进程上的统一度量时，我们证明了上述随机多项式$ f（x）$，对于某些常数$δ=δ（μ）> 0 $，概率$ \geΔ$是不可约的。实际上，如果算术进程具有第1步，我们证明了$ f（x）$的Galois组为$ \ MATHCAL {a} _n $或$ \ MATHCAL {S} _N $，具有概率$ \geδ$。
Let $μ$ be a probability measure on $\mathbb{Z}$ that is not a Dirac mass and that has finite support. We prove that if the coefficients of a monic polynomial $f(x)\in\mathbb{Z}[x]$ of degree $n$ are chosen independently at random according to $μ$ while ensuring that $f(0)\neq0$, then there is a positive constant $θ=θ(μ)$ such that $f(x)$ has no divisors of degree $\le θn$ with probability that tends to 1 as $n\to\infty$. Furthermore, in certain cases, we show that a random polynomial $f(x)$ with $f(0)\neq0$ is irreducible with probability tending to 1 as $n\to\infty$. In particular, this is the case if $μ$ is the uniform measure on a set of at least 35 consecutive integers, or on a subset of $[-H,H]\cap\mathbb{Z}$ of cardinality $\ge H^{4/5}(\log H)^2$ with $H$ sufficiently large. In addition, in all of these settings, we show that the Galois group of $f(x)$ is either $\mathcal{A}_n$ or $\mathcal{S}_n$ with high probability. Finally, when $μ$ is the uniform measure on a finite arithmetic progression of at least two elements, we prove a random polynomial $f(x)$ as above is irreducible with probability $\geδ$ for some constant $δ=δ(μ)>0$. In fact, if the arithmetic progression has step 1, we prove the stronger result that the Galois group of $f(x)$ is $\mathcal{A}_n$ or $\mathcal{S}_n$ with probability $\geδ$.
