Consider a semialgebraic function which is continuous around a point Using the so-called tangency variety of at we first provide necessary and sufficient conditions for to be a local minimizer of and then in the case where is an isolated local minimizer of we define a “tangency exponent” so that for any the following four conditions are always equivalent: (i) the inequality holds, (ii) the point is an th order sharp local minimizer of , (iii) the limiting subdifferential of is th order strongly metrically subregular at for 0, and (iv) the function satisfies the Łojaseiwcz gradient inequality at with the exponent Besides, we also present a counterexample to a conjecture posed by Drusvyatskiy and Ioffe [Math. Program. Ser. A, 153 (2015), pp. 635--653].