Proper polynomial mappings

Asked Mar 26 '18 at 21:00

Active Mar 27 '18 at 21:52

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One of the comments in this question says: "Polynomial mappings in more than one variable are generally not proper (definitely not when we work over $\mathbb{C}$) because the zero-sets are algebraic varieties and hence often unbounded (always over $\mathbb{C}$)."

(Proper map means preimages of compact sets are compacts).

(1) What happens over $\mathbb{R}$?

(2) Can one please elaborate on cases where such mappings are proper?

Thank you very much!

edited Mar 27 '18 at 21:52

Armando j18eos

3,907

asked Mar 26 '18 at 21:00

user237522

6,467

Consider the difference between $f(a,b) = a^2+b^2$ when defined as a map from $\mathbb{R}^2$ to $\mathbb{R}$ vs. $\mathbb{C}^2$ to $\mathbb{C}$, e.g. – Henno Brandsma Mar 27 '18 at 07:01
Thank you. Can you please elaborate? – user237522 Mar 27 '18 at 11:59
In the former case, fibres are circles in the plane (so compact) and a complex "circle" is unbounded, e.g. $(ni, n)$ is in the fibre of $0$ for all $n$ etc. – Henno Brandsma Mar 27 '18 at 12:12
Thanks for the explanation! Is it just a special case? – user237522 Mar 27 '18 at 12:34
Yes, $x^3 + y^3$ also has unbounded fibres over the reals but it shows how restricting to the reals sometimes does help. – Henno Brandsma Mar 27 '18 at 12:35
Thanks! This is very helpful. – user237522 Mar 27 '18 at 12:37

Proper polynomial mappings

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