Summary: Random polynomials with independent identically distributed Gaussian coefficients are considered. In the case of random gradient endomorphism \(F=(f,g): \mathbb R^2 \to \mathbb R^2\) the mean topological degree is computed and the expected number of complex points is estimated. In particular, the asymptotics of these invariants are determined as the algebraic degree of \(F\) tends to infinity. We also give the asymptotic of the mean writhing number of a standard equilateral random polygon with big number of sides.