Year: 2026
Type: article
Abstract: We study the proportion of conics given by $$(\mathcal {C}_{\varvec{F}, \varvec{y}}): F_0(\varvec{y})x_0^2 + F_1(\varvec{y})x_1^2 = F_2( \varvec{y})x_2^2 $$ ( C F , y ) : F 0 ( y ) x 0 2 + F 1 ( y ) x 1 2 = F 2 ( y ) x 2 2 which have a rational point $$\varvec{x} = (x_0:x_1:x_2) \in \mathbb {P}^2(\mathbb {Q})$$ x = ( x 0 : x 1 : x 2 ) ∈ P 2 ( Q ) , where $$\varvec{y} = (y_0: \dots : y_n)\in \mathbb {P}^n(\mathbb {Q})$$ y = ( y 0 : ⋯ : y n ) ∈ P n ( Q ) and $$F_0,F_1,F_2 \in \mathbb {Z}[X_0,\ldots , X_n]$$ F 0 , F 1 , F 2 ∈ Z [ X 0 , … , X n ] are homogeneous polynomials in many variables of the same degree d . We provide an asymptotic formula for the number of $$\varvec{y}$$
Source: Research in Number Theory
Author: Mathieu Da Silva
Institutions: Université Paris-Saclay, Laboratoire de Mathématiques d'Orsay
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,590
Funder: Université Paris-Saclay
OpenAlex ID: W7166540848