# Maximum modulus principle for vector valued functions of several complex variables

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)

• The paper lists a couple of conditions to be satisfied by the generic system (page 3). Commented Oct 8, 2022 at 17:01
• The statement is not correct: take $f(z,w)=z$ and $g(z)=\epsilon(z-1)$, then $(f,g)$ defines a polynomial map pf $C^2$ into itself, this map has no zeros, and $|f|^2+|g|^2$ has a minimum somewhere near the origin, when $\epsilon$ is small enough. Commented Oct 8, 2022 at 17:07