PreprintCC0arXiv
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≥1, a degree d form in n+1 variables, with independent uniform coefficients in {−1,1}, defines a singular complex hypersurface with probability On(d−1/2). The positive-dimensional singular loci occur with exponentially small probability. For n≥3, the same exponential bound holds for failure of absolute irreducibility. These results have been formalized in Lean by AxiomProver assuming existing literature.
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BibTeX
@misc{ono2026conjecture,
title = {{On a conjecture of Browning and Sawin on random hypersurfaces with sign coefficients}},
author = {Ken Ono and Ashvin Swaminathan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18879},
eprint = {2609.18879},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18879},
note = {CC0},
}
RIS
TY - UNPB
TI - On a conjecture of Browning and Sawin on random hypersurfaces with sign coefficients
AU - Ken Ono
AU - Ashvin Swaminathan
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18879
UR - https://arxiv.org/abs/2609.18879
AB - 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≥1, 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≥3, the same exponential bound holds for failure of absolute irreducibility. These results have been formalized in Lean by AxiomProver assuming existing literature.
ER -