Communicate Author or Publisher for asking detail? I have trouble finding out an expression in a theorem (click here to the problem)of a paper by Yann Bugeaud (page 12) , and I tried to contact the author through his e-mail but he didn't reply (don't know whether his e-mail is invalid or he ignored intentionally). 

I already asked this question on Math Stack Exchange without success. I have also posted it to math overflow, but it seems to me, it will not get much response.

Under the circumstances should I contact the publisher/editor of the paper or is there any other way to get the detail of the paper?
The editor and publisher will have no special insight into the contents of the paper.  The publisher doesn't even have any particular mathematical expertise at all.  It is a waste of time to contact either of them.

You speak about "the detail" as if you think that a more detailed version of the proof is out there somewhere, and if you ask the right person they will send it.  I think it's more likely that no such version ever exists.  The author may never have written another version - or if they did, there's no reason to think it exists anywhere besides their own notes.

Asking on MathOverflow was a good idea, and you can also talk to others in your field who may be familiar with the paper or the techniques involved.  But this is more a matter of having them help you figure out what's going on - not a matter of sharing knowledge that already exists.