It is well-known that if r is a rational number from [-1,0), then there is no polynomial f in two complex variables and a fiber $f^{-1}\left(t₀\right)$ such that r is the Łojasiewicz exponent of grad(f) near the fiber $f^{-1}\left(t₀\right)$. We show that this does not remain true if we consider polynomials in real variables. More exactly, we give examples showing that any rational number can be the Łojasiewicz exponent near the fiber of the gradient of some polynomial in real variables. The second main result of the paper is the formula...

For every polynomial F in two complex variables we define the Łojasiewicz exponents ${ℒ}_{p,t}\left(F\right)$ measuring the growth of the gradient ∇F on the branches centered at points p at infinity such that F approaches t along γ. We calculate the exponents ${ℒ}_{p,t}\left(F\right)$ in terms of the local invariants of singularities of the pencil of projective curves associated with F.