  The space $J^k$ of $k$-jets of a real function of one real variable $x$
admits the structure of Carnot group type. As such, $J^k$ admits a submetry
(\sR submersion) onto the Euclidean plane. Horizontal lifts of Euclidean lines
(which are the left-translates of horizontal one-parameter subgroups) are thus
globally minimizing geodesics on $J^k$. All $J^k$-geodesics, minimizing or not,
are constructed from degree $k$ polynomials in $x$ according to Anzaldo-Meneses
and Monroy-Per\'ez, reviewed here. The constant polynomials correspond to the
horizontal lifts of lines. Which other polynomials yield globally minimizers
and what do these minimizers look like? We give a partial answer. Our methods
include constructing an intermediate three-dimensional "magnetic"
sub-Riemannian space lying between the jet space and the plane, solving a
Hamilton-Jacobi (eikonal) equations on this space, and analyzing period
asymptotics associated to period degenerations arising from two-parameter
families of these polynomials. Along the way, we conjecture the independence of
the cut time of any geodesic on jet space from the starting location on that
geodesic.

Optimization and Control;Differential Geometry