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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)

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  • $\begingroup$ The paper lists a couple of conditions to be satisfied by the generic system (page 3). $\endgroup$
    – user125930
    Commented Oct 8, 2022 at 17:01
  • $\begingroup$ 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. $\endgroup$ Commented Oct 8, 2022 at 17:07

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