A complexity chasm for solving univariate sparse polynomial equations over $p$-adic fields

Abstract

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 $K\in \{{\mathbb{Q}}_2,{\mathbb{Q}}_3,{\mathbb{Q}}_5,\dots \}$, 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 $\Omega (d\log H)$ digits to distinguish the base-$p$ expansions of their roots in $K$.

19 pages, 3 figures. This version contains an Appendix missing from the ISSAC 2021 conference version, as well as some corrections and improvements

References in corpus (2)

• Zeros of sparse polynomials over local fields of characteristic p
• Bivariate trinomials over finite fields

Cited by in corpus (1)

• Sub-Linear Point Counting for Variable Separated Curves over Prime Power Rings