  This study examines the convexification version of the backward differential
flow algorithm for the global minimization of polynomials, introduced by O.
Arikan \textit{et al} in \cite{ABK}. It investigates why this approach might
fail with high-degree polynomials yet succeeds with quartic polynomials. We
employ the heat evolution method for convexification combined with Gaussian
filtering, which acts as a cumulative form of Steklov's regularization. In this
context, we apply the fingerprint theory from computer vision. Originally
developed by A.L. Yuille and T. Poggio in the 1980s for computer vision, the
fingerprint theory, particularly the fingerprint trajectory equation, is used
to illustrate the scaling (temporal) evolution of minimizers. In the case of
general polynomials, our research has led to the creation of the Yuille-Poggio
flow and a broader interpretation of the fingerprint concepts, in particular we
establish the condition both sufficient and necessary for the convexified
backward differential flow algorithms to successfully achieve global
minimization. For quartic polynomials, our analysis not only reflects the
results of O. Arikan et al. \cite{ABK} but also presents a significantly
simpler version of Newton's method that can always globally minimize quartic
polynomials without convexification.
