Łojasiewicz inequalities with explicit exponents for smallest singular value functions

Let F(x):=(fij(x))i=1,…,p;j=1,…,q, be a (p×q)-real polynomial matrix and let f(x) be the smallest singular value function of F(x). In this paper, we first give the following nonsmooth version of Łojasiewicz gradient inequality for the function f with an explicit exponent: For any x̄∈Rn, there exist c>0 and ϵ>0 such that we have for all ‖x−x̄‖<ϵ,inf{‖w‖:w∈∂f(x)}≥c|f(x)−f(x̄)|1−τ,where ∂f(x) is the limiting subdifferential of f at x, d:=maxi=1,…,p;j=1,…,qdegfij, ℛ(n,d):=d(3d−3)n−1 if d≥2 and ℛ(n,d):=1 if d=1, and τ:=1ℛ(n+p,2d+2). Then we establish some versions of Łojasiewicz inequality for the distance function with explicit exponents, locally and globally, for the smallest singular value function f(x) of the matrix F(x).