In the following paper: Shub and Smale, "On the Existence of Generally Convergent Algorithms", *Journal of Complexity* **2**, 2-11 (1986), trying to understand **Lemma 2** on page 4.

Paraphrased, below is the lemma:

Let $f: \mathbb{C}^n \rightarrow \mathbb{C}^n$ be a *generic* complex multivariate polynomial system. Define $g := \|f\|^2$, where the Hermitian inner product and norm are used. Let $\theta \in \mathbb{C}^n$ be a point satisfying $f(\theta) \neq 0$ and $\nabla{g} = 0$. Here $\nabla{g}$ is the gradient of $g$. Then $\theta$ is not a local minimum of $g$.

In the reference paper, the proof simply states that it is a consequence of the *maximum principle*. How exactly does the proof proceed in a little more detail?

What was tried thus far: tried adapting to this case the usual proof for the *minimum modulus principle* for holomorphic functions onto $\mathbb{C}$, where one considers the inverse function $1/f$ and then applies the maximum modulus principle - but that approach does not seem to directly apply since $f$ is vector-valued. Perhaps there is another way to prove this lemma? Of course, $g$ is a Morse function - does that help? (Note: just trying to better understand the paper - this is not a homework problem)