  We investigate the power of Algebraic Branching Programs (ABPs) augmented
with help polynomials, and constant-depth Boolean circuits augmented with help
functions. We relate the problem of proving explicit lower bounds in both these
models to the Remote Point Problem (introduced by Alon, Panigrahy, and Yekhanin
(RANDOM '09)). More precisely, proving lower bounds for ABPs with help
polynomials is related to the Remote Point Problem w.r.t. the rank metric, and
for constant-depth circuits with help functions it is related to the Remote
Point Problem w.r.t. the Hamming metric. For algebraic branching programs with
help polynomials with some degree restrictions we show exponential size lower
bounds for explicit polynomials.
