A global smooth version of the classical Łojasiewicz inequality

Let f:Rn→Rf:Rn→R be a function of class CdCd(d≥1)(d≥1) such that |∂df∂x1d|≥λ>0 on RnRn. Then the following global Łojasiewicz inequality holds true:λCddist(x,{f=0}∪{∂f∂x1=0})d≤|f(x)|for all x∈Rn, where Cd:=d!22d−1Cd:=d!22d−1 and dist(x,A)dist(x,A) denotes the Euclidean distance from x to A. As applications of this inequality, we have the following statements:
• If the sets {f=0}{f=0} and {∂f∂x1=0} are “non-asymptotic at infinity” then there exist positive constants ε and R such thatλCddist(x,{f=0})d≤|f(x)|whenever dist(x,{f=0})≤ε and ‖x‖≥R.
• If f is a polynomial of degree d with an isolated critical point at the origin, the following effective Łojasiewicz inequality holds truecdist(x,{f=0})d((2d−3)n+1)2≤|f(x)|for all ‖x‖≤r for some c>0c>0 and r>0r>0. Finally, we establish a relation between the above global Łojasiewicz inequality and the phenomenon of singularities at infinity.