(Revised) Does p(a)=p(b) => a=b?

Asked May 13 '13 at 12:32

Active May 13 '13 at 14:55

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This is a revised version of problem posted here named :Does $p(a) = p(b) \Rightarrow a=b \ $?

Let $S=(P_1,P_2,…,P_n)$ be a set of polynomials with complex coefficients.

I call S critical if it satisfies 2 conditions.

1.set of solutions of ($P_n(a)=P_n(b)\ \ |\forall n$) in $\Bbb C^2$ is (a=b) and

2.(ADDITIONAL CONDITION)(The subfield $K =\Bbb C(P_1(z),P_2(z) ,..,P_n(z))$ of $\Bbb C(z)$ does not contain z(i.e. K is a proper subfield))

I call a set minimal critical set if it is a critical set but no proper subset of it is critical.

Does There exist a minimal critical set?

if yes does there exist minimal critical sets of cardinality n for every n ?

What if we weaken condition ($a=b$) to {($a=b$) U (some finite points in $\Bbb C^2$)}? what about above questions? ((I think the set $(Z^2+Z,Z^4)$ works here but i am not sure , i dont know whether $\Bbb C(Z^2+Z, Z^4)= \Bbb C(Z)$or not) ),

edited Apr 13 '17 at 12:21

Community

asked May 13 '13 at 12:32

rohit

1

You should give a link to the previous version (and, if you intend to spend much time here, you should learn a little bit about formatting mathematics). – Gerry Myerson May 13 '13 at 12:42
Yep, my previous (deleted) comment was precisely because you've chosen odd notation to describe your condition. In particular, putting a $\forall$ at the end of a sentence made me miss it. – Thomas Andrews May 13 '13 at 12:48
@GerryMyerson Thanx for the comment, i gave the link for previous problem, For formatting where can i find some help? – rohit May 13 '13 at 14:36
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For some basic information about writing math at this site see e.g. here, here, here and here. – Lord_Farin May 13 '13 at 14:38
@Lord_Farin Thanx for help! – rohit May 13 '13 at 14:46

(Revised) Does p(a)=p(b) => a=b?

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