Non-Archimedean resonances of Łojasiewicz inequalities

We first present some survey about resonances of the Łojasiewicz inequalities in the realm of non-Archimedean geometry and their relation to Artin approximation. Next, we discuss the Łojasiewicz inequalities, along with some applications, established recently by the author in the settings of Hensel minimal structures with algebraic auxiliary sort RV. They come down, by means of our closedness theorem, to considering some induced sets which are definable in a Presburger language in the value group sort; and eventually, to certain problems of piecewise linear geometry over the field of rational numbers. In other words, the problems in question have been reduced to study the piecewise linear shadows of definable sets, being objects with a polyhedral structure.