We reveal a complexity chasm, separating the trinomial and tetranomial cases, for solving univariate sparse polynomial equations over certain local fields. First, for any fixed field $K\in\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5,\ldots\}$, we prove that any polynomial $f\in\mathbb{Z}[x_1]$ with exactly $3$ monomial terms, degree $d$, and all coefficients having absolute value at most $H$, can be solved over $K$ in deterministic time $\log^{O(1)}(dH)$ in the classical Turing model. (The best previous algorithms were of complexity exponential in $\log d$, even for just counting roots in $\mathbb{Q}_p$.) In particular, our algorithm generates approximations in $\mathbb{Q}$ with bit-length $\log^{O(1)}(dH)$ to all the roots of $f$ in $K$, and these approximations converge quadratically under Newton iteration. On the other hand, we give a unified family of {\em tetra}nomials requiring $\Omega(d\log H)$ bits to distinguish the base-$b$ expansions of their roots in $K$.

中文翻译：

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求解 $p$-adic 域上的稀疏多项式方程的复杂性鸿沟

我们揭示了一个复杂的鸿沟，将三项式和四项式情况分开，用于求解特定局部域上的单变量稀疏多项式方程。首先，对于任意固定域 $K\in\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5,\ldots\}$，我们证明任意多项式 $f\in\mathbb {Z}[x_1]$ 具有正好 $3$ 的单项式项，度数 $d$，并且所有系数的绝对值最多为 $H$，可以在确定性时间内通过 $K$ 求解 $\log^{O(1) }(dH)$ 在经典图灵模型中。（以前最好的算法在 $\log d$ 中的复杂度呈指数级，即使只是计算 $\mathbb{Q}_p$ 中的根也是如此。）特别是，我们的算法在 $\mathbb{Q}$ 中生成近似值长度 $\log^{O(1)}(dH)$ 到 $K$ 中 $f$ 的所有根，并且这些近似值在牛顿迭代下二次收敛。另一方面，