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In this paper, we investigate the possibility of thebackward-differential-flow-like algorithm which starts from the minimum ofconvexification version of the polynomial. We apply the heat evolutionconvexification approach through Gaussian filtering, which is actually anaccumulation version of Steklov's regularization. We generalize the fingerprinttheory which was proposed in the theory of computer vision by A.L. Yuille andT. Poggio in 1980s, in particular their fingerprint trajectory equation, tocharacterize the evolution of minimizers across the scale. On the other hand,we propose the "seesaw" polynomials $p(x|s)$ and we find a seesaw differentialequation $\frac{\partial p(x|s)}{\,ds}=-\frac{1}{p''(x)}$ to characterize theevolution of global minimizer $x^*(s)$ of $p(x|s)$ while varying $s$.Essentially, both the fingerprints $\mathcal{FP}_2$ and $\mathcal{FP}_3$ of$p(x)$, consisting of the zeros of $\frac{\partial^2 p(x,t)}{\partial x^2}$ and$\frac{\partial^3 p(x,t)}{\partial x^3}$, respectively, are independent ofseesaw coefficient $s$, upon which we define the Confinement Zone and EscapeZone. Meanwhile, varying $s$ will monotonically condition the location ofglobal minimizer of $p(x|s)$, and all these location form the Attainable Zone.Based on these concepts, we prove that the global minimizer $x^*$ of $p(x)$ canbe inversely evolved from the global minimizer of its convexificationpolynomial $p(x,t_0)$ if and only if $x^*$ is included in the Escape Zone. Inparticular, we give detailed analysis for quartic and six degree polynomials.

Yuille-Poggio's Flow and Global Minimizer of polynomials through convexification by Heat Evolution