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

Abstract: We study the proportion of conics given by $$(\mathcal{C}<em>{\mathbf{F}, \mathbf{y}}) : F_0(\mathbf{y})x_0² + F_1(\mathbf{y})x_1² = F_2( \mathbf{y})x_2²$$ which have a rational point $$\mathbf{x} = (x_0 :x_1:x_2) \in \mathbb{P}<sup>2(\mathbb{Q})$$, where $$\mathbf{y} = (y_0 : \dots : y_n)\in \mathbb{P}<sup>n(\mathbb{Q})$$ and $F_0,F_1,F_2 \in \mathbb{Z}[X_0,\ldots, X_n]$ 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 $(\mathcal{C}</em>{\mathbf{F}, \mathbf{y}})$ 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 $$t_0x_0² + t_1x_1² + t_2x_2² = 0$$ having a rational point, and coefficients $t_0,t_1,t_2$ in arithmetic progressions.