Strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices

Using the variational method, it is shown that the set of all strong peak functions in a closed algebra $A$ of $C_b(K)$ is dense if and only if the set of all strong peak points is a norming subset of $A$. As a corollary we can induce the denseness of strong peak functions on other certain spaces. In case that a set of uniformly strongly exposed points of a Banach space $X$ is a norming subset of $\mathcal{P}({}^n X)$, then the set of all strongly norm attaining elements in $\mathcal{P}({}^n X)$ is dense. In particular, the set of all points at which the norm of $\mathcal{P}({}^n X)$ is Fr\'echet differentiable is a dense $G_\delta$ subset. In the last part, using Reisner's graph theoretic-approach, we construct some strongly norm attaining polynomials on a CL-space with an absolute norm. Then we show that for a finite dimensional complex Banach space $X$ with an absolute norm, its polynomial numerical indices are one if and only if $X$ is isometric to $\ell_\infty^n$. Moreover, we give a characterization of the set of all complex extreme points of the unit ball of a CL-space with an absolute norm.