Let f: ℝⁿ → ℝ be a C² semialgebraic function and let c be an asymptotic critical value of f. We prove that there exists a smallest rational number ${\varrho }_c\le 1$ such that |x|·|∇f| and ${|f\left(x\right)-c|}^{{\varrho }_c}$ are separated at infinity. If c is a regular value and ${\varrho }_c<1$, then f is a locally trivial fibration over c, and the trivialisation is realised by the flow of the gradient field of f.

An effective formula for the Łojasiewicz exponent for analytic curves in a neighbourhood of 0 ∈ ℂ is given.

The Łojasiewicz exponent of the gradient of a convergent power series h(X,Y) with complex coefficients is the greatest lower bound of the set of λ > 0 such that the inequality ${\left|gradh(x,y)\right|\ge c\left|(x,y)\right|}^{\lambda }$ holds near $0\in C^2$ for a certain c > 0. In the paper, we give an estimate of the Łojasiewicz exponent of grad h using information from the Newton diagram of h. We obtain the exact value of the exponent for non-degenerate series.