Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Ono, Ken, Swaminathan, Ashvin
Institution: Palo Alto University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213523357