  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.
