关于多个变量中两个不同次数的型的系统
On two forms in many variables of different degrees
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中文总结 AI 辅助
本文研究多个变量中两个不同次数的型的整数解，在奇异轨迹维数较小且次数差足够大时，通过新差分论证结合 van der Corput 方法，显著降低了渐近公式成立的变量个数阈值，改进了先前结果。
中文摘要 AI 辅助
本文研究满足多个变量中两个不同次数的型的系统的整数解。设 $d_1$ 和 $d_2$ 为自然数，且 $d_2>d_1\geq 2$。设 $F_{i}(\boldsymbol{x}) \\ (i=1,2)$ 分别为 $n$ 个变量中次数为 $d_i\\ (i=1,2)$ 的型。定义 $$N(\boldsymbol{F};P):=\\#\{\boldsymbol{x}\in [-P,P]^n\cap \mathbb{Z}^n:\\ F_{i}(\boldsymbol{x})=0\\ (i=1,2)\}.$$ 当 $F_1=0$ 和 $F_2=0$ 的奇异轨迹的各维数较小时，我们得到一个数 $n_0:=n_0(\boldsymbol{F})$，使得只要 $n> n_0$，就有预期的渐近公式 \begin{equation*} N(\boldsymbol{F};P)=c_{\boldsymbol{F}}\cdot P^{n-d_1-d_2}+O(P^{n-d_1-d_2-\delta}),\\ \text{对某个 }\delta>0, \end{equation*} 其中常数 $c_{\boldsymbol{F}}$ 是局部密度的乘积。我们注意到该渐近公式与 Manin-Peyre 猜想一致。与先前的工作相比，在大多数情况下我们降低了可容许阈值 $n_0(\boldsymbol{F})$，但 $d_2-d_1=1$ 的情形除外。特别地，若 $F_1$ 和 $F_2$ 是非奇异型，则在 $d_2\geq 5d_1$ 且 $d_1\geq2$ 的条件下，我们得到 $$n_0(\boldsymbol{F})=3(d_2-1)2^{d_2-1}+(d_1-1)2^{d_1},$$ 这比先前的界 $n_0(\boldsymbol{F})=(d_1+2)(d_2-1)2^{d_2-1}+d_12^{d_1-1}$ 有显著改进。为实现这一目标，我们发展了一种新的差分论证，并结合 van der Corput 差分论证，当次数差足够大时，为与多个变量中两个不同次数的型相关的指数和均值提供了有效的上界估计。此外，本文所述方法具有足够的灵活性，可推广应用于一般情形下多个变量中次数不同的型。
英文摘要
In this paper, we investigate integral solutions satisfying the system of two forms in many variables of different degrees. Let $d_1$ and $d_2$ be natural numbers with $d_2>d_1\geq 2$. Let $F_{i}(\boldsymbol{x}) \ (i=1,2)$ be forms in $n$ variables of degrees $d_i\ (i=1,2)$, respectively. Define $$N(\boldsymbol{F};P):=\#\{\boldsymbol{x}\in [-P,P]^n\cap \mathbb{Z}^n:\ F_{i}(\boldsymbol{x})=0\ (i=1,2)\}.$$ When each dimension of singular loci of $F_1=0$ and $F_2=0$ is small, we obtain a number $n_0:=n_0(\boldsymbol{F})$ such that whenever $n> n_0$ one has the expected asymptotic formula \begin{equation*} N(\boldsymbol{F};P)=c_{\boldsymbol{F}}\cdot P^{n-d_1-d_2}+O(P^{n-d_1-d_2-δ}),\ \text{for some }δ>0, \end{equation*} where the constant $c_{\boldsymbol{F}}$ is the product of local densities. We note that this asymptotic formula agrees with the Manin-Peyre conjecture. Compared to the previous work, we lower the admissible threshold $n_0(\boldsymbol{F})$ in most cases, with the exception of case $d_2-d_1=1$. In particular, if $F_1$ and $F_2$ are non-singular forms, then we obtain $$n_0(\boldsymbol{F})=3(d_2-1)2^{d_2-1}+(d_1-1)2^{d_1},$$ provided that $d_2\geq 5d_1$ with $d_1\geq2$. This yields a substantial improvement over the previous bound $n_0(\boldsymbol{F})=(d_1+2)(d_2-1)2^{d_2-1}+d_12^{d_1-1}$. To achieve this, we develop a new differencing argument together with the van der Corput differencing argument, delivering an efficient upper-bound estimate for mean values of exponential sums associated with two forms in many variables of different degrees, when the difference between degrees is sufficiently large. Furthermore, the method described in this paper is flexible enough to apply to forms in many variables of differing degrees in general.
发表机构
- Department of Mathematics, University of California, Davis（加州大学戴维斯分校数学系）
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