The local solubility for homogeneous polynomials with random coefficients over thin sets
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| Format: | Preprint |
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2023
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| _version_ | 1866914027797479424 |
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| author | Lee, Heejong Lee, Seungsu Yeon, Kiseok |
| author_facet | Lee, Heejong Lee, Seungsu Yeon, Kiseok |
| contents | Let $d$ and $n$ be natural numbers greater or equal to $2$. Let $\langle \boldsymbol{a}, ν_{d,n}(\boldsymbol{x})\rangle\in \mathbb{Z}[\boldsymbol{x}]$ be a homogeneous polynomial in $n$ variables of degree $d$ with integer coefficients $\boldsymbol{a}$, where $\langle\cdot,\cdot\rangle$ denotes the inner product, and $ν_{d,n}: \mathbb{R}^n\rightarrow \mathbb{R}^N$ denotes the Veronese embedding with $N=\binom{n+d-1}{d}$. Consider a variety $V_{\boldsymbol{a}}$ in $\mathbb{P}^{n-1}$, defined by $\langle \boldsymbol{a}, ν_{d,n}(\boldsymbol{x})\rangle=0.$ In this paper, we examine a set of these varieties defined by
$$\mathbb{V}^{P}_{d,n}(A)=\{ V_{\boldsymbol{a}}\subset \mathbb{P}^{n-1}|\ P(\boldsymbol{a})=0,\ \|\boldsymbol{a}\|_{\infty}\leq A\},$$
where $P\in \mathbb{Z}[\boldsymbol{x}]$ is a non-singular form in $N$ variables of degree $k$ with $2 \le k\leq C({n,d})$ for some constant $C({n,d})$ depending at most on $n$ and $d$.
Suppose that $P(\boldsymbol{a})=0$ has a nontrivial integer solution. We confirm that the proportion of varieties $V_{\boldsymbol{a}}$ in $\mathbb{V}^{P}_{d,n}(A)$, which are everywhere locally soluble, converges to a constant $c_P$ as $A\rightarrow \infty.$ In particular, if there exists $\boldsymbol{b}\in \mathbb{Z}^N$ such that $P(\boldsymbol{b})=0$ and the variety $V_{\boldsymbol{b}}$ in $\mathbb{P}^{n-1}$ admits a smooth $\mathbb{Q}$-rational point, the constant $c_P$ is positive. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_13685 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The local solubility for homogeneous polynomials with random coefficients over thin sets Lee, Heejong Lee, Seungsu Yeon, Kiseok Number Theory Algebraic Geometry Let $d$ and $n$ be natural numbers greater or equal to $2$. Let $\langle \boldsymbol{a}, ν_{d,n}(\boldsymbol{x})\rangle\in \mathbb{Z}[\boldsymbol{x}]$ be a homogeneous polynomial in $n$ variables of degree $d$ with integer coefficients $\boldsymbol{a}$, where $\langle\cdot,\cdot\rangle$ denotes the inner product, and $ν_{d,n}: \mathbb{R}^n\rightarrow \mathbb{R}^N$ denotes the Veronese embedding with $N=\binom{n+d-1}{d}$. Consider a variety $V_{\boldsymbol{a}}$ in $\mathbb{P}^{n-1}$, defined by $\langle \boldsymbol{a}, ν_{d,n}(\boldsymbol{x})\rangle=0.$ In this paper, we examine a set of these varieties defined by $$\mathbb{V}^{P}_{d,n}(A)=\{ V_{\boldsymbol{a}}\subset \mathbb{P}^{n-1}|\ P(\boldsymbol{a})=0,\ \|\boldsymbol{a}\|_{\infty}\leq A\},$$ where $P\in \mathbb{Z}[\boldsymbol{x}]$ is a non-singular form in $N$ variables of degree $k$ with $2 \le k\leq C({n,d})$ for some constant $C({n,d})$ depending at most on $n$ and $d$. Suppose that $P(\boldsymbol{a})=0$ has a nontrivial integer solution. We confirm that the proportion of varieties $V_{\boldsymbol{a}}$ in $\mathbb{V}^{P}_{d,n}(A)$, which are everywhere locally soluble, converges to a constant $c_P$ as $A\rightarrow \infty.$ In particular, if there exists $\boldsymbol{b}\in \mathbb{Z}^N$ such that $P(\boldsymbol{b})=0$ and the variety $V_{\boldsymbol{b}}$ in $\mathbb{P}^{n-1}$ admits a smooth $\mathbb{Q}$-rational point, the constant $c_P$ is positive. |
| title | The local solubility for homogeneous polynomials with random coefficients over thin sets |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2308.13685 |