  In this letter, we prove the existence of resummed expressions for the
diagonal of the heat kernel and the effective action of a quantum field which
interacts with a scalar or an electromagnetic background. Working in an
arbitrary number of spacetime dimensions, we propose an Ansatz beyond the
Schwinger--DeWitt proposal, effectively resumming an infinite number of
invariants which can be constructed from powers of the background, as well as
its first and second derivatives in the Yukawa case. This provides a proof of
the recent conjecture that all terms containing the invariants
$F_{\mu\nu}F^{\mu\nu}$ and $\widetilde F_{\mu\nu}F^{\mu\nu}$ in the proper-time
series expansion of the SQED effective action can be resummed. Possible
generalizations and several applications are also discussed -- in particular,
the existence of an analogue of the Schwinger effect for Yukawa couplings.
