Three parameters of Boolean functions related to their constancy on affine spaces

<p style='text-indent:20px;'>The <inline-formula><tex-math id="M1">\begin{document}$ k $\end{document}</tex-math></inline-formula>-normality of Boolean functions is an important notion initially introduced by Dobbertin and studied in several papers. The parameter related to this notion is the maximal dimension of those affine spaces contained in the support <inline-formula><tex-math id="M2">\begin{document}$ supp(f) $\end{document}</tex-math></inline-formula> of the function or in its co-support <inline-formula><tex-math id="M3">\begin{document}$ cosupp(f) $\end{document}</tex-math></inline-formula>. We denote it by <inline-formula><tex-math id="M4">\begin{document}$ norm\,(f) $\end{document}</tex-math></inline-formula> and call it the norm of <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula>. <p style='text-indent:20px;'>The norm concerns only the affine spaces contained in either the support or the co-support; the information it provides on <inline-formula><tex-math id="M6">\begin{document}$ f $\end{document}</tex-math></inline-formula> is then somewhat incomplete (for instance, two functions constant on a hyperplane will have the same very large parameter value, while they can have very different complexities). A second parameter which completes the information given by the first one is the minimum between the maximal dimension of those affine spaces contained in <inline-formula><tex-math id="M7">\begin{document}$ supp(f) $\end{document}</tex-math></inline-formula> and the maximal dimension of those contained in <inline-formula><tex-math id="M8">\begin{document}$ cosupp(f) $\end{document}</tex-math></inline-formula> (while <inline-formula><tex-math id="M9">\begin{document}$ norm\,(f) $\end{document}</tex-math></inline-formula> equals the maximum between these two maximal dimensions). We denote it by <inline-formula><tex-math id="M10">\begin{document}$ cons\,(f) $\end{document}</tex-math></inline-formula> and call it the (affine) constancy of <inline-formula><tex-math id="M11">\begin{document}$ f $\end{document}</tex-math></inline-formula>. <p style='text-indent:20px;'>The value of <inline-formula><tex-math id="M12">\begin{document}$ cons\,(f) $\end{document}</tex-math></inline-formula> gives global information on <inline-formula><tex-math id="M13">\begin{document}$ f $\end{document}</tex-math></inline-formula>, but no information on what happens around each point of <inline-formula><tex-math id="M14">\begin{document}$ supp(f) $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M15">\begin{document}$ cosupp(f) $\end{document}</tex-math></inline-formula>. We define then its local version, equal to the minimum, when <inline-formula><tex-math id="M16">\begin{document}$ a $\end{document}</tex-math></inline-formula> ranges over <inline-formula><tex-math id="M17">\begin{document}$ \Bbb{F}_2^n $\end{document}</tex-math></inline-formula>, of the maximal dimension of those affine spaces which contain <inline-formula><tex-math id="M18">\begin{document}$ a $\end{document}</tex-math></inline-formula> and on which <inline-formula><tex-math id="M19">\begin{document}$ f $\end{document}</tex-math></inline-formula> is constant. We denote it by <inline-formula><tex-math id="M20">\begin{document}$ stab\,(f) $\end{document}</tex-math></inline-formula> and call it the stability of <inline-formula><tex-math id="M21">\begin{document}$ f $\end{document}</tex-math></inline-formula>. <p style='text-indent:20px;'>We study the properties of these three parameters. We have <inline-formula><tex-math id="M22">\begin{document}$ norm\,(f)\geq cons\,(f)\geq stab\,(f) $\end{document}</tex-math></inline-formula>, then for determining to which extent these three parameters are distinct, we exhibit four infinite classes of Boolean functions, which show that all cases can occur, where each of these two inequalities can be strict or large. <p style='text-indent:20px;'>We consider the minimal value of <inline-formula><tex-math id="M23">\begin{document}$ stab\, (f) $\end{document}</tex-math></inline-formula> (resp. <inline-formula><tex-math id="M24">\begin{document}$ cons\,(f) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M25">\begin{document}$ norm\,(f) $\end{document}</tex-math></inline-formula>), when <inline-formula><tex-math id="M26">\begin{document}$ f $\end{document}</tex-math></inline-formula> ranges over the Reed-Muller code <inline-formula><tex-math id="M27">\begin{document}$ RM(r,n) $\end{document}</tex-math></inline-formula> of length <inline-formula><tex-math id="M28">\begin{document}$ 2^n $\end{document}</tex-math></inline-formula> and order <inline-formula><tex-math id="M29">\begin{document}$ r $\end{document}</tex-math></inline-formula>, and we denote it by <inline-formula><tex-math id="M30">\begin{document}$ stab\, _{RM(r,n)} $\end{document}</tex-math></inline-formula> (resp. <inline-formula><tex-math id="M31">\begin{document}$ cons\, _{RM(r,n)} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M32">\begin{document}$ norm\, _{RM(r,n)} $\end{document}</tex-math></inline-formula>). We give upper bounds for each of these three integer sequences, and determine the exact values of <inline-formula><tex-math id="M33">\begin{document}$ stab\, _{RM(r,n)} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M34">\begin{document}$ cons\, _{RM(r,n)} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M35">\begin{document}$ r\in\{1,2,n-2,n-1,n\} $\end{document}</tex-math></inline-formula>, and of <inline-formula><tex-math id="M36">\begin{document}$ norm\, _{RM(r,n)} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M37">\begin{document}$ r = 1,2 $\end{document}</tex-math></inline-formula>.