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

Mathieu Da Silva

TL;DR

The paper analyzes the solubility of a family of conics $(m C_{m F,m y})$ parametrized by $m y otinoldsymbol Q$ with coefficients given by homogeneous polynomials $F_i$ of degree $d$ in many variables, and proves an asymptotic count for the number of $m y$ with height up to $B$ for which the conic has a rational point. Employing a circle-method strategy rooted in the work of Destagnol–Lyczak–Sofos, together with a Selberg–Delange analysis for multi-variable arithmetic functions, the authors establish a Birch-system–based asymptotic $N_{ ext{glob}}(oldsymbol F,B)\, hicksim\, c(oldsymbol F) rac{B}{(\log B)^{3/2}}$. The constant $c(oldsymbol F)$ is computed via a detailed study of the subordinate Brauer group, local densities, and Tamagawa-type measures, yielding explicit product formulas and a demonstration that the expected Loughran–Rome–Sofos-type density governs the count. The diagonal-conic analysis and AP-coefficient results serve as a crucial intermediate input to verify the constant predictions and to implement the circle-method framework effectively. The findings support the LRS conjectures in this setting and provide explicit local-global density data for the family of conics. The work advances the understanding of rational-solubility proportions in high-dimensional polynomial families and provides a blueprint for deriving exact constants in similar fibration problems.

Abstract

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 $\mathbf{y} = (y_0 : \dots : y_n)\in \mathbb{P}^n(\mathbb{Q})$ 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 $(\mathcal{C}_{\mathbf{F}, \mathbf{y}})$ 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 $t_0x_0^2 + t_1x_1^2 + t_2x_2^2 = 0$ having a rational point, and coefficients $t_0,t_1,t_2$ in arithmetic progressions.

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

TL;DR

The paper analyzes the solubility of a family of conics parametrized by with coefficients given by homogeneous polynomials of degree in many variables, and proves an asymptotic count for the number of with height up to for which the conic has a rational point. Employing a circle-method strategy rooted in the work of Destagnol–Lyczak–Sofos, together with a Selberg–Delange analysis for multi-variable arithmetic functions, the authors establish a Birch-system–based asymptotic . The constant is computed via a detailed study of the subordinate Brauer group, local densities, and Tamagawa-type measures, yielding explicit product formulas and a demonstration that the expected Loughran–Rome–Sofos-type density governs the count. The diagonal-conic analysis and AP-coefficient results serve as a crucial intermediate input to verify the constant predictions and to implement the circle-method framework effectively. The findings support the LRS conjectures in this setting and provide explicit local-global density data for the family of conics. The work advances the understanding of rational-solubility proportions in high-dimensional polynomial families and provides a blueprint for deriving exact constants in similar fibration problems.

Abstract

We study the proportion of conics given by which have a rational point , where and are homogeneous polynomials in many variables of the same degree . We provide an asymptotic formula for the number of of bounded height such that the corresponding conic 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 having a rational point, and coefficients in arithmetic progressions.

Paper Structure

This paper contains 20 sections, 30 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Motivations and first definitions.
  3. The Loughran--Smeets conjecture.
  4. Main result.
  5. Prediction for the constant
  6. The subordinate Brauer group of $\pi_{\bm F}$.
  7. Prediction for the constant.
  8. Application of the Selberg--Delange method
  9. Proportion of diagonal plane conics with a rational point and coefficients in arithmetic progression
  10. Step 1 : preliminary calculations.
  11. Step 2 : estimating $\mathcal{N}_{\bm b,\bm m,\bm r}({\bm X})$.
  12. Step 3 : estimating $\mathcal{M}_{\bm b,\bm m,\bm r}({\bm X})$.
  13. Step $4$: returning to $\mathcal{N}_{\bm b,\bm m,\bm r}({\bm X})$.
  14. Step 5 : returning to $N'({\bm n},q,B)$
  15. Computing the constant
  16. ...and 5 more sections

Key Result

Theorem 1.7

Let $d \geqslant 1$ and $F_0,F_1,F_2 \in \mathbb{Z}[X_0, \ldots, X_n]$ be three homogeneous polynomials of degree $d$, forming a Birch system as defined in Definition polynomes de birch. Assume that for all $i \in \{0,1,2\}$, the varieties $V_i$ defined by the equation $F_i = 0$ are smooth and that where $c(\pi_{\bm F})$ is the constant (cst conj) predicted by den.

Theorems & Definitions (63)

  • Definition 1.1
  • Definition 1.2
  • Conjecture 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 1.6
  • Theorem 1.7
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • ...and 53 more