Enumerative Galois theory for cubics and quartics
Enumerative Galois theory for cubics and quartics
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DOI：
10.1016/j.aim.2020.107282
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发表时间：
2020-10
影响因子：
1.7
通讯作者：
Sam Chow;R. Dietmann
中科院分区：
文献类型：
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作者：
Sam Chow;R. Dietmann
证明了存在O ε（H1.5 + ε）一次、三次多项式，其Galois群为A3，其整系数绝对值有界于H.我们还证明了D4四次数的数量级为H2（log H）2，A4，V4，C4的相应计数为O（H2.91），O（H2 log H），O（H2 log H）.我们的工作建立了不可约的非S 3三次多项式是为数不多的比可约的，同样在四次设置：这是前两个解决的情况下，1936年猜想由货车德尔Waerden。
We show that there are O ε (H 1.5+ ε) monic, cubic polynomials with integer coefficients bounded by H in absolute value whose Galois group is A 3. We also show that the order of magnitude for D 4 quartics is H 2 (log H) 2, and that the respective counts for A 4, V 4, C 4 are O (H 2.91), O (H 2 log H), O (H 2 log H). Our work establishes that irreducible non-S 3 cubic polynomials are less numerous than reducible ones, and similarly in the quartic setting: these are the first two solved cases of a 1936 conjecture made by van der Waerden.