The paper describes several algorithms related to a problem of computing the local dimension of a semialgebraic set. Let a semialgebraic set V be defined by a system of k inequalities of the form f โฅ 0 with f โ R[X 1 , . . . , Xn], deg(f ) < d, and x โ V . An algorithm is constructed for computing the dimension of the Zariski tangent space to V at x in time (kd) O(n) . Let x belong to a stratum of codimension lx in V with respect to a smooth stratification of V . Another algorithm computes the local dimension dimx(V ) with the complexity (k(lx + 1)d) O(l 2
x n) . If l = max xโV lx, and for every connected component the local dimension is the same at each point, then the algorithm computes the dimension of every connected component with complexity (k(l + 1)d) O(l 2 n) . If V is a real algebraic variety defined by a system of equations, then the complexity of the algorithm is less than kd O(l 2 n) , and the algorithm also finds the dimension of the tangent space to V at x in time kd O(n) . When l is fixed, like in the case of a smooth V , the complexity bounds for computing the local dimension are (kd) O(n) and kd O(n) respectively. A third algorithm finds the singular locus of V in time (kd) O(n 2 ) .