This post is a sequel of: [When should a supervisor be a co-author?][1]     
 
This time the topic is about the interaction between two professional mathematicians (in particular junior-senior, but not necessarily).    

Of course, this will depend on the nature of the interaction (Q&A on a specific subject or talks or informal discussions), its frequency (1 time/year or /month or /week or /day) and also on the level of the mathematicians: what can look a high contribution for one, can look not so high for an other. Also, someone don't need or just don't want to be a co-author, because it's not enough high level for him.        
I've also heard about senior mathematicians of high level, allowing several discussions on a specific joint work with a lot of contributors, and then becomes co-author, just by writing a nice introduction, but without having written any detailed proof: like a conductor and his musicians.   

Anyway, the mathematicians interact during conferences, by emails, on mathoverflow or in  their institute.    
   
**Q:**  How to distinguish the interactions which should lead to a collaboration  or to an acknowledgment?  
____  
>A general answer may be a proper *adoption* of the American patent law rejection (of a coautorship): the alleged research contribution by the coauthor does not *rise to the dignity of research*.   
 

Thus this would be the central question here. The general issue is the ethical quality of life in the creative environment (science, inventions, art, ...). Here we concentrate on mathematical research publications. Arguably, the research credit is in this case among the most vital issues, including cases of being prevented from a publication. The mathematical credit is given (or should be) for theories, theorems, conjectures, definitions, even notation. Smooth expositions and monographs are highly valued (and regularly win academic advencement for the authors) but they are not strictly research. Now about more specific questions (an invitation to Answer and Comment):

**An acknowledgement within a paper: what is ethical? (examples)**

 1. inside the regular text;
 2. a formal ***Acknowledgement*** at the end of Introduction or the whole paper;
 3. inside ***Introduction***;
 4. credit for using an unpublished result of one of the coauthors.



**Giving credit among the coauthors (in their paper): what is (not) reasonable?**

 1. The order of listing the coauthors (mathematical default: alphabetic; there are exceptions);
 2. Coauthors are discrete about the division of credit among them (mathematical default) or sometimes they spell out each author's contribution.



**Kleptomania and stealing (never mind acceptable, but how to prevent it?)**


More questions can be raised or some may be even erased.

>**An opinion** (wh): a publication should adequately represent the research contributions of all involved, both of (co)authors and by others (non-coauthors). The word is ***adequately***, without unnecessary details or assigning any weight to the researchers. A coauthor whose **research** contribution does not rise to the dignity of research should not be a coauthor.

  [1]: https://mathoverflow.net/q/57337/34538