Mathematics Colloquia and Seminars

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Let $d$ and $n$ be natural numbers greater or equal to $2$. Let $\langle \boldsymbol{a}, \nu_{d,n}(\boldsymbol{x})\rangle\in \mathbb{Z}[\boldsymbol{x}]$ 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 $\nu_{d,n}: \mathbb{R}^n\rightarrow \mathbb{R}^N$ denotes the Veronese embedding with $N=\binom{n+d-1}{d}$. Consider a variety $V_{\boldsymbol{a}}$ in $\mathbb{P}^{n-1}$, defined by $\langle \boldsymbol{a}, \nu_{d,n}(\boldsymbol{x})\rangle=0.$ In this talk, we examine a set of integer vectors $\boldsymbol{a}\in \mathbb Z^N$, defined by $$\mathfrak{A}(A;P)=\{ \boldsymbol{a}\in \mathbb Z^N:\ P(\boldsymbol{a})=0,\ \|\boldsymbol{a}\|_{\infty}\leq A\},$$ 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 integer vectors $\boldsymbol{a}\in \mathbb Z^N$ in $\mathfrak{A}(A)$, whose associated equation $\langle \boldsymbol{a}, \nu_{d,n}(\boldsymbol{x})\rangle=0$ is everywhere locally soluble, converges to a constant $c_P$ as $A\rightarrow \infty.$ Moreover, for each place $v$ of $\mathbb Q$, if there exists a non-zero $\boldsymbol{b}_v\in \mathbb Q_v^N$ such that $P(\boldsymbol{b}_v)=0$ and the variety $V_{\boldsymbol{b}_v}$ in $\mathbb{P}^{n-1}$ admits a smooth $\mathbb{Q}_v$-point, the constant $c_P$ is positive. This is joint work with Heejong Lee and Seungsu Lee.