论文标题
在高度值的频率上
On the frequency of height values
论文作者
论文摘要
我们将固定度$ D $的代数数和固定度（绝对乘法Weil）高度$ \ MATHCAL {H} $与完全位于开放单元磁盘内的$ k $ conjugates进行计数。我们还计算了最多$ \ MATHCAL {H} $的值，该高度假定为$ d $的代数数量$ d $，精确地位于开放单元磁盘内。对于这两项计数，我们都不获得渐近的生长，而仅获得粗糙的生长顺序，这是由于计数函数对数的渐近生长而产生的。对于第一个计数，即使是$ k \ in \ {0，d \} $或$ \ gcd（k，d）= 1 $，也存在这种粗糙的增长顺序。因此，我们研究了$ 0 <k <d $和$ \ gcd（k，d）> 1 $的情况。我们还计算固定度和固定Mahler测量的整数多项式，并用固定数量的开放单元磁盘（以多重性计数）进行固定数量的复合物零件，并研究高度函数的动力学行为。
We count algebraic numbers of fixed degree $d$ and fixed (absolute multiplicative Weil) height $\mathcal{H}$ with precisely $k$ conjugates that lie inside the open unit disk. We also count the number of values up to $\mathcal{H}$ that the height assumes on algebraic numbers of degree $d$ with precisely $k$ conjugates that lie inside the open unit disk. For both counts, we do not obtain an asymptotic, but only a rough order of growth, which arises from an asymptotic for the logarithm of the counting function; for the first count, even this rough order of growth exists only if $k \in \{0,d\}$ or $\gcd(k,d) = 1$. We therefore study the behaviour in the case where $0 < k < d$ and $\gcd(k,d) > 1$ in more detail. We also count integer polynomials of fixed degree and fixed Mahler measure with a fixed number of complex zeroes inside the open unit disk (counted with multiplicities) and study the dynamical behaviour of the height function.
