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I'm an absolute beginner in this matter, but I want to know the following:

If I understand correctly, once a Math paper is accepted for publication in a journal, they will ask you to submit the LaTeX source file. Once you send that file, does that journal work on the code and edit it in order to correct any technical inefficiencies (commands, spaces, alignment, etc.)?

After this, do they send you the corrected LaTeX code for you to see it before it is used for publication?

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I've never received LaTeX source code from a publisher. I've received proofs (i.e. printed version) to review but not the source code. Publishers typically have their own styles and, even if they make a copy available to you, they will do some "tweaking".

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And sometimes they won't. It varies wildly. – mbork Mar 31 '12 at 11:23

I had been in the publishing field for last 12 years. As far as I know authors source file (Latex or any other type) will not be edited or disturbed. The Publishers/Typesetter will rename the authors source file for copyediting and the copyedited file will be paginated. Authors will be supplied with the typeset proof for review (all the errors in the source file would have been fixed during editing and Quality check processes). However, author will be queried if the source file is unusable or has any missing style files!!! Also, publishers will request authors to submit a PDF version of the manuscript as a reference copy. Hope this clears your doubt!!!

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It didn't happen to me with an important publisher: my paper (written in quite a good LaTeX, I dare say) was completely retyped, introducing the largest number of misprints and misunderstandings of math symbol I've ever found. – egreg Apr 2 '12 at 5:59
Same here. At least 10-15 years ago when I was still publishing things it was usual to send hardcopies to india for retyping. From what I know now about this business, this trend is getting even stronger. – Stephan Lehmke Apr 2 '12 at 6:37

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