The author studies the Łojasiewicz gradient exponent at infinity for polynomials in two variables. First he obtains an effective estimate for critical values of a polynomial in one variable [cf. S. Smale, Bull. Am. Math. Soc., New Ser. 4, 1–36 (1981; Zbl 0456.12012)] and generalizes the Bochnak-Łojasiewicz inequality from S. Spodzieja [Bull. Pol. Acad. Sci., Math. 50, No. 3, 273–281 (2002; Zbl 1057.14076)] as follows: for a polynomial map \(f \colon \mathbb C^n \rightarrow \mathbb C\) there are constants \(C, \varepsilon > 0\) such that the condition \(| f(z)| \leq \varepsilon\) provides \(| z| |\text{grad}\, f(z)| \geq C| f(z)|.\) Then he considers a polynomial in two variables of degree \(d >2\) such that \(0\in \mathbb C\) is a bifurcation point of \(f\) at infinity and proves that for any neighbourhood of the fiber \(f^{-1}(0)\) the gradient exponent of \(f\) does not exceed \((-1-1/(d-2)).\) In particular, this implies a sharper version of Malgrange condition \((m)\) [A.Parusinski, Compos. Math. 97, No. 3, 369–384 (1995; Zbl 0840.32007)]. The author also proves the following separation condition for a polynomial in two variables and its gradient: \(| f(z)| \leq \varepsilon \Rightarrow |\text{grad}\, f(z)| \geq C| f(z)|^q\) for suitable \(C, \varepsilon, q >0.\)