We are interested in how small a root of multiplicity k
can be for a power series of the form f(z:=
1+[sum ]∞n=1aizi
with coefficients ai in
[−1, 1]. Let r(k) denote the size of
the smallest root of multiplicity k
possible for such a power series. We show that
formula here
We describe the form that the extremal power series must take
and develop an algorithm that lets us
compute the optimal root (which proves to be an algebraic number).
The computations, for k[les ]27, suggest
that the upper bound is close to optimal and that
r(k)∼1−c/(k+1),
where c=1.230….