  We show that for every homogeneous polynomial of degree $d$, if it has
determinantal complexity at most $s$, then it can be computed by a homogeneous
algebraic branching program (ABP) of size at most $O(d^5s)$. Moreover, we show
that for $\textit{most}$ homogeneous polynomials, the width of the resulting
homogeneous ABP is just $s-1$ and the size is at most $O(ds)$.
  Thus, for constant degree homogeneous polynomials, their determinantal
complexity and ABP complexity are within a constant factor of each other and
hence, a super-linear lower bound for ABPs for any constant degree polynomial
implies a super-linear lower bound on determinantal complexity; this relates
two open problems of great interest in algebraic complexity. As of now,
super-linear lower bounds for ABPs are known only for polynomials of growing
degree, and for determinantal complexity the best lower bounds are larger than
the number of variables only by a constant factor.
  While determinantal complexity and ABP complexity are classically known to be
polynomially equivalent, the standard transformation from the former to the
latter incurs a polynomial blow up in size in the process, and thus, it was
unclear if a super-linear lower bound for ABPs implies a super-linear lower
bound on determinantal complexity. In particular, a size preserving
transformation from determinantal complexity to ABPs does not appear to have
been known prior to this work, even for constant degree polynomials.
