Timeline for How many values a polynomial map misses?
Current License: CC BY-SA 3.0
13 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jan 29, 2013 at 18:49 | comment | added | dwan | Thomas Scanlon: I do not see how Podewski's conjecture implies an answer to my question. My confusion is that for my field F, I usually have a polynomial f such that its value set f(F) (a definable subset of F) is neither finite nor co-finite and thus the field F is not minimal. For example, as you mention that Podewski's conjecture is true in positive characteristic. What this implies for my value gap question when F is an infinite algebraic extension of a finite field, the case of most interest to me. Can you explain in more detail? | |
| Jan 29, 2013 at 10:53 | comment | added | HJRW | Thomas Scanlon - it seems to me you could post your comment as an answer. | |
| Jan 29, 2013 at 6:18 | comment | added | Thomas Scanlon | This question is a special case of a problem which has been open for four decades: Is an infinite field $K$ for which every definable set $X \subseteq K$ is finite or cofinite [called a ``minimal field''] necessarily algebraically closed? See Podewski, Minimale Ringe, Math. Phys. Semesterber 22 (1975), no. 2, 193 – 197. It is known that a counterexample must have characteristic zero: Wagner, Minimal fields, J. Symbolic Logic 65 (2000), no. 4, 1833 – 1835. | |
| Jan 29, 2013 at 6:03 | history | edited | dwan | CC BY-SA 3.0 |
updated with a multi-variable version.
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| Jan 29, 2013 at 5:59 | comment | added | dwan | I have updated this question with a multi-variate version, which seems newer, even in the case when the field F is the complex numbers. | |
| Jan 29, 2013 at 5:58 | history | edited | dwan | CC BY-SA 3.0 |
added 22 characters in body; edited title
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| Jan 29, 2013 at 5:50 | history | edited | dwan | CC BY-SA 3.0 |
added 676 characters in body; added 195 characters in body
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| Jan 29, 2013 at 5:27 | comment | added | Greg Kuperberg | Note that when I asked a related question, Bjorn gave a non-rigorous argument that there should be counterexamples. I think that "no actual answer" is too pessimistic; one can conjecture with evidence that the answer is no. mathoverflow.net/questions/9863/… | |
| Jan 29, 2013 at 5:06 | comment | added | Noam D. Elkies | So should this be closed as essentially a duplicate of MO6820? | |
| Jan 29, 2013 at 4:29 | comment | added | Gjergji Zaimi | This question has appeared before with no actual answer mathoverflow.net/questions/6820/… | |
| Jan 29, 2013 at 4:21 | history | edited | dwan |
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| Jan 29, 2013 at 4:15 | history | asked | dwan | CC BY-SA 3.0 |