{"ID":22964817,"CreatedAt":"2026-09-17T02:29:35.102872001Z","UpdatedAt":"2026-09-17T02:29:35.102872001Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.18879","arxiv_id":"2609.18879","title":"On a conjecture of Browning and Sawin on random hypersurfaces with sign coefficients","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.","short_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...","url_abs":"https://arxiv.org/abs/2609.18879","url_pdf":"https://arxiv.org/pdf/2609.18879v1","authors":"[\"Ken Ono\",\"Ashvin Swaminathan\"]","published":"2026-09-16T16:14:25Z","proceeding":"math.NT","tasks":"[\"math.NT\",\"math.AG\"]","methods":"[]","has_code":false}