Solubility of a family of conics with polynomial coefficients in many variables

2025/11/25 by Da Silva, Mathieu
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2511.20282

openalex publication_date 2025/11/25 · openalex created_date 2025/11/28 · openalex updated_date 2026/07/28

Abstract

We study the proportion of conics given by (C_{F, y}) : F₀(y)x₀² + F₁(y)x₁² = F₂( y)x₂² which have a rational point x = (x₀ :x₁:x₂) ∈ ℙ²(ℚ), where y = (y₀ : … : y_n)∈ ℙⁿ(ℚ) and F₀,F₁,F₂ ∈ ℤ[X₀,…, X_n] are homogeneous polynomials in many variables of the same degree d. We provide an asymptotic formula for the number of y of bounded height such that the corresponding conic (C_{F, y}) has a rational point. In particular, our result agrees with the Loughran--Smeets and the Loughran--Rome--Sofos conjectures. Our strategy is based on a recent result of Destagnol--Lyczak--Sofos relying on the circle method to estimate the average of an arithmetic function over polynomials in many variables. To this end, we study the proportion of conics t₀x₀² + t₁x₁² + t₂x₂² = 0 having a rational point, and coefficients t₀,t₁,t₂ in arithmetic progressions.

Citations

• Local solubility of generalised Fermat equations
• Local solubility of a family of ternary conics over a biprojective base I
• Asymptotics For Local Solubility Of Diagonal Quadrics Over A Split Quadric Surface
• The leading constant for rational points in families
• The density of fibres with a rational point for a fibration over hypersurfaces of low degree
• Fibrations with few rational points
• The number of varieties in a family which contain a rational point
• Manin's conjecture for certain biprojective hypersurfaces
• Hauteurs et mesures de Tamagawa sur les variétés de Fano

Related

• Local solubility of a family of ternary conics over a biprojective base I
• The leading constant for rational points in families
• Local solubility in generalised Châtelet varieties
• Solubility of a resultant equation and applications
• Higher Rédei reciprocity and integral points on conics