Maurice J. Rojas

## Institution:

Texas A&M University

## Time:

Tuesday, January 7, 2020 - 10:00am to 10:50am

## Location:

RH 306

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.