论文标题
解决单变量稀疏多项式方程的复杂性鸿沟超过$ p $ -
A complexity chasm for solving univariate sparse polynomial equations over $p$-adic fields
论文作者
论文摘要
我们揭示了复杂性的鸿沟，分隔了三项和四型病例，用于求解某些局部场上的单变量稀疏多项式方程。首先，对于任何固定字段$ k \ in \ {\ mathbb {q} _2，\ mathbb {q} _3，\ mathbb {q} _5，\ ldots \} $，我们证明，我们证明，任何一个多项式$ f \ in \ mathbb {z}绝对值最多可以在确定时间$ o（\ log^{o（1）}（dh）（dh））$中在经典图灵模型中求解$ k $。 (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 $O(\log^{O(1)}(dH))$ to all the roots of $f$ in $K$, and these在牛顿迭代下，近似二次收敛。另一方面，我们给出了一个统一的四元家庭，需要$ω（d \ log h）$数字，以区分其根源以$ k $为基础。
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 $f\in\mathbb{Z}[x]$ with exactly $3$ monomial terms, degree $d$, and all coefficients having absolute value at most $H$, can be solved over $K$ in deterministic time $O(\log^{O(1)}(dH))$ 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 $O(\log^{O(1)}(dH))$ 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 tetranomials requiring $Ω(d\log H)$ digits to distinguish the base-$p$ expansions of their roots in $K$.
