Metric Properties of Semialgebraic Mappings

We give an effective estimation from above for the local Łojasiewicz exponent for separation of semialgebraic sets and for a semialgebraic mapping on a closed semialgebraic set. We also give an effective estimation from below of the Łojasiewicz exponent in the global separation for semialgebraic sets and estimation of the Łojasiewicz exponent at infinity of a semialgebraic mapping similar to the Jelonek result [14] in the complex case. Moreover, we prove that both local and global Łojasiewicz exponent of an overdetermined semialgebraic mapping F : X → R on a closed semialgebraic set X ⊂ R (i.e. m > dimX) are equal to the Łojasiewicz exponent of the composition L ◦ F : X → R for the generic linear mapping L : R → R, where k = dimX.