具有指定系数的整数多项式的伽罗瓦群
Galois groups of integer polynomials with prescribed coefficients
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中文总结 AI 辅助
本文研究固定部分系数的整数多项式伽罗瓦群，证明当自由系数足够多时，非对称群例外计数为H^{s-1}阶，结合原始群筛与稀疏Hermite插值，并处理低次端点情形。
中文摘要 AI 辅助
固定次数 $n$、非零整数常数项以及一个首一整数多项式的若干其他系数，留下 $s$ 个系数在 $[-H,H]$ 中变化。我们首先证明，当仅固定首项和常数项时，对于每个 $n\ge4$，其伽罗瓦群不是 $S_n$ 的多项式数量为 $H^{n-2}$ 阶。然后我们证明一个半维扩展：对于 $n\ge6$，当 $s>n/2+1$ 时，相应的例外计数为 $H^{s-1}$ 阶，前提是系数切片以指定的维数界满足高重数轨迹。如果自由系数形成初始或最终块，则此几何条件对每个指定值成立，并且对任意系数位置的非空 Zariski 开集指定值也成立。它还对每个额外指定的内部系数值成立。因此，在次数七和八中，除了首项和常数项外，还可以固定任意一个内部系数。证明结合了 Bhargava 原始群筛的固定范数版本与稀疏 Hermite 插值。一个单独的非本原估计，使用最小中间域，在有理部分积超曲面之外给出 $O(H^{\beta(n)})$，其中对于具有最小素因子 $\ell$ 的合数 $n$，$\beta(n)=\ell+n/\ell-2\le n/2$，且没有对数损失。固定常数的四次和五次端点需要单独的局部论证：$A_4$ 四次多项式的例外 Fourier 方向和二次 Gauss 和用于五次。我们进一步分离偶次端点结果，解释不同的三次障碍，并总结在 Malle 猜想的上界形式下的条件改进。
英文摘要
Fix a degree $n$, a nonzero integer constant term, and some further coefficients of a monic integer polynomial, leaving $s$ coefficients to vary in $[-H,H]$. We first prove that, when only the leading and constant coefficients are fixed, the number of polynomials whose Galois group is not $S_n$ is of order $H^{n-2}$ for every $n\ge4$. We then prove a half-dimensional extension: for $n\ge6$ the corresponding exceptional count is of order $H^{s-1}$ whenever $s>n/2+1$, provided the coefficient slice meets the higher-multiplicity loci with a specified dimension bound. This geometric condition holds for every prescribed value if the free coefficients form an initial or final block, and for a nonempty Zariski-open set of prescribed values for arbitrary coefficient positions. It also holds for every value of one additional prescribed interior coefficient. Thus, in degrees seven and eight, any one interior coefficient can be fixed in addition to the leading and constant coefficients. The proof combines a fixed-norm version of Bhargava's primitive-group sieve with sparse Hermite interpolation. A separate imprimitive estimate, using a minimal intermediate field, gives $O(H^{β(n)})$ outside rational partial-product hypersurfaces, where $β(n)=\ell+n/\ell-2\le n/2$ for composite $n$ with smallest prime divisor $\ell$, and has no logarithmic loss. The fixed-constant quartic and quintic endpoints require separate local arguments: exceptional Fourier directions for $A_4$ quartics and a quadratic Gauss sum for quintics. We isolate further even-degree endpoint results, explain the distinct cubic obstruction, and conclude with conditional improvements under the upper-bound form of Malle's conjecture.
