  The study of inner and cyclic functions in $\ell^p_A$ spaces requires a
better understanding of the zeros of the so-called optimal polynomial
approximants.
  We determine that a point of the complex plane is the zero of an optimal
polynomial approximant for some element of $\ell^p_A$ if and only if it lies
outside of a closed disk (centered at the origin) of a particular radius which
depends on the value of $p$. We find the value of this radius for $p\neq 2$. In
addition, for each positive integer $d$ there is a polynomial $f_d$ of degree
at most $d$ that minimizes the modulus of the root of its optimal linear
polynomial approximant. We develop a method for finding these extremal
functions $f_d$ and discuss their properties. The method involves the Lagrange
multiplier method and a resulting dynamical system.
