Let \(f:\mathbb{R}^{2}\rightarrow \mathbb{R}\) be a polynomial mapping with a finite number of critical points (i.e. points \((x,y)\in \mathbb{R}^{2}\) such that \(\text{grad}f(x,y)=(0,0))\). For every \(p\in \mathbb{R}^{2}\) (or \( p=\infty \)) let \(i_{p}(f)\) be the topological degree of \(\text{grad}f\) at \(p\) (it is the topological degree of the mapping \(\text{grad}f/\left| \text{grad}f\right| :S(p,\varepsilon )\rightarrow S(0,1),\) where \( \varepsilon \ll 1\) for \(p\) finite and \(\varepsilon \gg 1\) for \(p=\infty \)) and \(r_{p}(f)\) be the number of branches of the curve \(f(x,y)=f(p)\) at \(p\). Arnold proved \(i_{p}(f)=1-r_{p}(f)\) for \(p\in \mathbb{R}^{2}\) and M. Sȩkalski [Ann. Pol. Math. 87, 229–235 (2005; Zbl 1091.14017)] proved a similar result for \( p=\infty \). The main result of the paper is the formula \[ i_{\infty }(f)=1+\int_{\mathbb{R}}r_{\infty }(t)d\chi (t), \] where \(r_{\infty }(t)\) denotes the number of branches at infinity of the curve \(f(x,y)=t\) and \(\int_{\mathbb{R}}\varphi (t)d\chi (t)\) is the Euler integral of \(\varphi \).