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For any fixed field $K\!\in\!\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5,\ldots\}$, we prove that all polynomials $f\!\in\!\mathbb{Z}[x]$ with exactly$3$ (resp. $2$) monomial terms, degree $d$, and all coefficients havingabsolute value at most $H$, can be solved over $K$ within deterministic time$\log^{7+o(1)}(dH)$ (resp. $\log^{2+o(1)}(dH)$) in the classical Turing model:Our underlying algorithm correctly counts the number of roots of $f$ in $K$,and for each such root generates an approximation in $\mathbb{Q}$ withlogarithmic height $O(\log^3(dH))$ that converges at a rate of$O\!\left((1/p)^{2^i}\right)$ after $i$ steps of Newton iteration. We alsoprove significant speed-ups in certain settings, a minimal spacing bound of$p^{-O(p\log^2_p(dH)\log d)}$ for distinct roots in $\mathbb{C}_p$, and evenstronger repulsion when there are nonzero degenerate roots in $\mathbb{C}_p$:$p$-adic distance $p^{-O(\log_p(dH))}$. On the other hand, we prove that thereis an explicit family of tetranomials with distinct nonzero roots in$\mathbb{Z}_p$ indistinguishable in their first $\Omega(d\log_p H)$ mostsignificant base-$p$ digits.

Root Repulsion and Faster Solving for Very Sparse Polynomials Over $p$-adic Fields