A complexity chasm for solving univariate sparse polynomial equations over p-adic fields
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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∈{ℚ_2,ℚ_3,ℚ_5,…}, we prove that any polynomial f∈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 ℚ_p.) In particular, our algorithm generates approximations in 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(dlog H) digits to distinguish the base-p expansions of their roots in K.
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