Łojasiewicz inequalities with explicit exponents for smallest singular value functions

Let $F(x) := (f_{ij}(x))_{i=1,\ldots,p; j=1,\ldots,q},$ be a ($p\times q$)-real polynomial matrix and let $f(x)$ be the smallest singular value function of $F(x).$ In this paper, we first give the following {\em nonsmooth} version of \L ojasiewicz gradient inequality for the function $f$ with an explicit exponent: {\em For any $\bar x\in \Bbb R^n$, there exist $c > 0$ and $\epsilon > 0$ such that we have for all $\|x - \bar{x}\| < \epsilon,$ \begin{equation*} \inf \{ \| w \| \ : \ w \in {\partial} f(x) \} \ \ge \ c\, |f(x)-f(\bar x)|^{1 - \frac{2}{\mathscr R(n+p,2d+2)}}, \end{equation*} where ${\partial} f(x)$ is the limiting subdifferential of $f$ at $x$, $d:=\max_{i=1,\ldots,p; j=1,\ldots,q}\deg f_{i j}$ and $\mathscr R(n, d) := d(3d - 3)^{n-1}$ if $d \ge 2$ and $\mathscr R(n, d) := 1$ if $d = 1.$} Then we establish some versions of \L ojasiewicz inequality for the distance function with explicit exponents, locally and globally, for the smallest singular value function $f(x)$ of the matrix $F(x)$.