A note on singularities at infinity of complex polynomials

Let f be a complex polynomial. We relate the behaviour of f "at infinity" to the sheaf of vanishing cycles of the family ${\displaystyle \overline{f}}$ of projective closures of fibres of f. We show that the absence of such cycles: (i) is equivalent to a condition on the asymptotic behaviour of gradient of f known as Malgrange's Condition, (ii) implies the ${\displaystyle C^{\infty }}$-triviality of f. If the support of sheaf of vanishing cycles of ${\displaystyle \overline{f}}$ is a finite set, then it detects precisely the change of the topology of the fibres of f. Moreover, in this case, the generic fibre of f has the homotopy type of a bouquet of spheres.