arXiv - CS - Computational Complexity Pub Date : 2020-02-29 , DOI: arxiv-2003.00314
J. Maurice Rojas; Yuyu Zhu

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 $K$.

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