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

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