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

We study the proportion of conics given by $({\mathcal{C}}_{\mathbf{F},\mathbf{y}}):F_0(\mathbf{y})x_0^2+F_1(\mathbf{y})x_1^2=F_2(\mathbf{y})x_2^2$ which have a rational point $\mathbf{x}=(x_0:x_1:x_2)\in {\mathbb{P}}^2(\mathbb{Q})$, where $\mathbf{y}=(y_0:\cdots :y_n)\in {\mathbb{P}}^n(\mathbb{Q})$ and $F_0,F_1,F_2\in \mathbb{Z}[X_0,\dots ,X_n]$ are homogeneous polynomials in many variables of the same degree $d$. We provide an asymptotic formula for the number of $\mathbf{y}$ of bounded height such that the corresponding conic $({\mathcal{C}}_{\mathbf{F},\mathbf{y}})$ 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 $t_0{x}_0^2+t_1{x}_1^2+t_2{x}_2^2=0$ having a rational point, and coefficients $t_0,t_1,t_2$ in arithmetic progressions.