For $1\leq k \leq m,$ let $f_k :\mathbb{C}^n \rightarrow \mathbb{C}$ be a multivariate polynomial map. With $\mathbf{x} \equiv (x_1,\ldots,x_n)$, consider the map $F(\mathbf{x}) \equiv \{f_1(\mathbf{x}), f_2(\mathbf{x}), \ldots, f_m(\mathbf{x}) \},$ where $m > n.$ Is it true that $Im(F)$ has measure zero in $\mathbb{C}^m.$

  • $\begingroup$ I want an overdetermined system. So $m > n$ $\endgroup$ – Suresh Apr 1 '12 at 7:37

Yes. Since each $F$ is locally Lipschitz, the image has Hausdorff dimension $\le n$.

  • $\begingroup$ You mean dimension is less than equal to n? $\endgroup$ – Suresh Apr 1 '12 at 8:02
  • $\begingroup$ Sorry, yes. I'll edit. $\endgroup$ – Robert Israel Apr 1 '12 at 17:31

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