Milnor number of positive polynomials (Q1399083)

The total Milnor number of a complex polynomial \(f : \mathbb C^n \to \mathbb C \) with isolated affine singularities is known to be equal to the sum of the local affine Milnor numbers. The author shows the following theorem: Let \(f\) be a complex polynomial with real coefficients and let the restriction of \(f\) to \(\mathbb R^n\) be bounded from below and moreover let the norm of the gradient of \(f\) be bounded away from zero for \(\| z\|\) sufficiently large, then the total Milnor number of \(f\) is odd. The proof is short. It uses curve selection lemma and a Morsification.