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How many digits of an algebraic number A do you need to know before you

are sure you know A? This question dates back to the early 20th century (if

not earlier) and work of Kurt Mahler on the minimal spacing between

complex roots of a degree d univariate polynomial f with integer coefficients of

absolute value at most h: One can bound the minimal spacing explicitly as a function

of d and h. However, the optimality of Mahler's bound for sparse polynomials was open

until recently.

We give a unified family of examples, having just 4 monomial terms, showing

Mahler's bound to be asyptotically optimal over both the p-adic complex numbers,

and the usual complex numbers. However, for polynomials with 3

or fewer terms, we show how to significantly improve Mahler's bound, in both

the p-adic and Archimedean cases. As a consequence, we show how certain

sparse polynomials of degree d can be ``solved'' in time (log d)^{O(1)} over certain local fields.

This is joint work with Yuyu Zhu.