关于Browning和Sawin关于带符号系数随机超曲面的一个猜想的证明
On a conjecture of Browning and Sawin on random hypersurfaces with sign coefficients
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中文总结 AI 辅助
本文证明了Browning和Sawin关于带符号系数随机超曲面光滑性的猜想，给出了奇异概率的定量界，并指出正维奇异轨迹概率指数小，结果已在Lean中形式化。
中文摘要 AI 辅助
Browning和Sawin猜想：当次数趋于无穷时，带符号系数的随机超曲面是光滑的概率趋于1。我们证明了这一猜想，并得到了一个定量界。对于每个$n\geq1$，在$n+1$个变量中，系数独立且均匀取自$\{-1,1\}$的$d$次型，定义奇异复超曲面的概率为$O_n(d^{-1/2})$。正维奇异轨迹出现的概率呈指数小。对于$n\geq3$，同样的指数界也适用于绝对不可约性失败的情况。这些结果已由AxiomProver在Lean中形式化，假设现有文献成立。
英文摘要
Browning and Sawin conjectured that random hypersurfaces with sign coefficients are smooth with probability tending to one as the degree grows. We prove this conjecture and obtain a quantitative bound. For each $n\geq1$, a degree $d$ form in $n+1$ variables, with independent uniform coefficients in $\{-1,1\}$, defines a singular complex hypersurface with probability $O_n(d^{-1/2})$. The positive-dimensional singular loci occur with exponentially small probability. For $n\geq3$, the same exponential bound holds for failure of absolute irreducibility. These results have been formalized in Lean by AxiomProver assuming existing literature.
发表机构
- Axiom Math（Axiom数学研究院）
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