论文标题
三个变量中的二次一致性小溶液的渐近行为模型素
Asymptotic behavior of small solutions of quadratic congruences in three variables modulo prime powers
论文作者
论文摘要
令$ p> 5 $为固定的素数，并假设$α_1，α_2，α_3$是企业与$ p $。 We study the asymptotic behavior of small solutions of congruences of the form $α_1x_1^2+α_2x_2^2+α_3x_3^2\equiv 0\bmod{q}$ with $q=p^n$, where $\max\{|x_1|,|x_2|,|x_3|\}\le N$ and $（x_1x_2x_3，p）= 1 $。 （实际上，我们考虑了此问题的平滑版本。）如果$α_1，α_2，α_3$是固定的，并且$ n \ rightarrow \ infty $，我们建立了一个渐近公式（因此存在此类解决方案）$ n \ gg q^{1/2+\ varepsilon} $。如果允许这些系数随$ n $变化，我们表明，如果$ n \ gg q^{11/18+\ varepsilon} $，则该公式成立。后者应与Heath-Brown的结果进行比较，Heath-Brown在条件下建立了非零解决方案的存在，用于$ n \ gg q^{5/8+\ varepsilon} $，用于奇数无正方形的模量$ q $。
Let $p>5$ be a fixed prime and assume that $α_1,α_2,α_3$ are coprime to $p$. We study the asymptotic behavior of small solutions of congruences of the form $α_1x_1^2+α_2x_2^2+α_3x_3^2\equiv 0\bmod{q}$ with $q=p^n$, where $\max\{|x_1|,|x_2|,|x_3|\}\le N$ and $(x_1x_2x_3,p)=1$. (In fact, we consider a smoothed version of this problem.) If $α_1,α_2,α_3$ are fixed and $n\rightarrow \infty$, we establish an asymptotic formula (and thereby the existence of such solutions) under the condition $N\gg q^{1/2+\varepsilon}$. If these coefficients are allowed to vary with $n$, we show that this formula holds if $N\gg q^{11/18+\varepsilon}$. The latter should be compared with a result by Heath-Brown who established the existence of non-zero solutions under the condition $N \gg q^{5/8+\varepsilon}$ for odd square-free moduli $q$.
