  We study extensions of piecewise polynomial data prescribed on faces and
possibly in elements of a patch of simplices sharing a vertex. In the $H^1$
setting, we look for functions whose jumps across the faces are prescribed,
whereas in the ${\bf H}(\mathrm{div})$ setting, the normal component jumps and
the piecewise divergence are prescribed. We show stability in the sense that
the minimizers over piecewise polynomial spaces of the same degree as the data
are subordinate in the broken energy norm to the minimizers over the whole
broken $H^1$ and ${\bf H}(\mathrm{div})$ spaces. Our proofs are constructive
and yield constants independent of the polynomial degree. One particular
application of these results is in a posteriori error analysis, where the
present results justify polynomial-degree-robust efficiency of potential and
flux reconstructions.
