论文标题
单基因纯立方体
Monogenic pure cubics
论文作者
论文摘要
令$ k \ geq 2 $为无平方的整数。我们证明了[1，n] $ in [1，n] $中的无正式整数$ m \，以至于$（k，m）= 1 $和$ \ mathbb {q}}（\ sqrt [3] {k^2m}）$是$ \ gg n^{1/3} $ and $ \ ll n/（ $ε> 0 $。假设ABC可以将上限提高到$ o（n^{（1/3）+ε}）$。令$ f $为$ q $的有限字段，$（q，3）= 1 $，让$ g（t）\在f [t] $中为非稳定方形。我们无条件地证明了类似的结果，即F [t] $中的无平方$ H（t）\的数量，使得$°（h）\ leq n $，$（g，h）= 1 $ and $ f（t，\ sqrt [3] {g^2h}）$是$ \ gg q q^q^^n/3 ll} n^2q^{n/3} $。
Let $k\geq 2$ be a square-free integer. We prove that the number of square-free integers $m\in [1,N]$ such that $(k,m)=1$ and $\mathbb{Q}(\sqrt[3]{k^2m})$ is monogenic is $\gg N^{1/3}$ and $\ll N/(\log N)^{1/3-ε}$ for any $ε>0$. Assuming ABC, the upper bound can be improved to $O(N^{(1/3)+ε})$. Let $F$ be the finite field of order $q$ with $(q,3)=1$ and let $g(t)\in F[t]$ be non-constant square-free. We prove unconditionally the analogous result that the number of square-free $h(t)\in F[t]$ such that $°(h)\leq N$, $(g,h)=1$ and $F(t,\sqrt[3]{g^2h})$ is monogenic is $\gg q^{N/3}$ and $\ll N^2q^{N/3}$.
