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First page of On a conjecture of Browning and Sawin on random hypersurfaces with sign coefficients

On a conjecture of Browning and Sawin on random hypersurfaces with sign coefficients

Ken Ono, Ashvin Swaminathan

math.NT Sep 16, 2026 · v1 math.AG
formalization algebra number-theory
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TL;DR
The main results on smoothness and irreducibility of random sign hypersurfaces were formalized in Lean by AxiomProver assuming existing literature.
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.
Problem

Browning and Sawin conjectured that random hypersurfaces with sign coefficients in {-1,1} become smooth with probability tending to one as the degree grows. The task is to prove this and give a quantitative bound.

Approach

The singular locus is reduced to a finite-field problem via reduction modulo a prime, using the Jacobian criterion and Poonen-style local sieves. Concentration estimates for sign sums (via linear independence and Fourier-Hölder arguments) bound singularity probabilities at closed points across degree ranges. The prime and cutoff are chosen comparable to sqrt(d) and log d respectively to obtain the final complex-number bounds. The main theorems were formalized in Lean by AxiomProver assuming existing literature.

Results

For each n≥1, a degree-d sign form in n+1 variables defines a singular complex hypersurface with probability O_n(d^{-1/2}), proving the conjecture. Positive-dimensional singular loci and, for n≥3, failure of absolute irreducibility occur with exponentially small probability.