Non-mathematician submitting to top maths journal? I am an amateur mathematician, and I had an idea which I worked out a bit and sent to an expert. He urged me to write it up for publication. So I did, and put it on arXiv. There were a couple of rounds of constructive criticism, after which he tells me he thinks it ought to go a "top" journal (his phrase, and he went on to name two). 
His opinion is that my outsider status will have no effect on the reviewing process and my paper will be taken seriously. I am pleased and quite flattered to hear this, and my inclination is to do as he suggests. But I have to say it sounds too good to be true. Does this match other people's experience? I understand it's rare enough for undergraduates to publish anywhere, and I am not even an undergraduate! Surely a "non-mathematician submitting to top mathematical journal" must instantly rank high on the crackpot scale. How often does this actually occur successfully these days? Any advice?
 A: Think of it this way: any graduate student who submits his first result for publication faces a similar issue: his paper will land on a referee's desk, who has never heard the name of the author before, so according to your logic, the better the result the more it must smell of crackpottery to the referee. Yet, PhD students and young postdocs do get their results published and in fact, they sometimes receive more patience from the referee, when it comes to the writing style, say, than experienced mathematicians.
Fact is, that an expert in the respective discipline can recognise 99% of the crackpots within less than 10 seconds. If your write-up passes this 10 second test and looks like a serious attempt at a mathematical problem, then the referee will read it. Of course, he might be extra sceptical about actual mistakes and extra diligent, but that can only be good for you.
At any rate, the threshold to receiving a careful reading is not nearly as high as you think. Key to this is a well written abstract and introduction with clear and precise statements of the results and, ideally, a clear indication of where the main difficulties were and how you overcame them, i.e. where the actual novelty lies. That's "all" it takes (I am not claiming that it's easy, but obtaining the result in the first place was likely harder).
And one last word: receiving rejections is part of the business. A mathematician who cannot handle them will lead a miserable life. None of us likes rejections, but we all must contend with the possibility of getting one. If you think that you will not be able to digest a rejection, then send the paper to a less prestigious journal. After all, your mental well-being is more important than this kind of prestige, especially since you are not relying on getting things published.
A: What do you have to lose by submitting an article for publication?  You'll have an even better record/credentialing/verification of the work you've put into it by being published in a Journal of good reputation.  In the worst case, you will get a rejection letter, perhaps with a good explanation of why they are rejecting the article.  The in-between case is pretty good too: you'll receive a referee report which may criticize your approach, suggest particular points to be polished and corrected, perhaps suggest a different approach to take, perhaps suggest that some of this work has been done by others who you shoud read and study or perhaps refer to in your work.
If you're lucky enough to be asked to rewrite and resubmit for consideration, you're possibly on your way to being published.  If not, you'll at least have made some progress and educated yourself about the academic publication culture, and will be more prepared for the next article which you prepare for submission.
May I recommend that you find out what your activation energy level is, exceed it, and go ahead and clean up your paper and submit it for publication.  Best wishes and good luck.  Go for it!
Would you mind sharing the arxiv link to your work?
A: First of all, congratulations for making it as far as you have. I have a similar story, in 1986 I met with Stephen Wolfram and discussed with him my research. Wolfram was very supportive and even offered to edit and publish my research if I would just write it up. Then Wolfram left academia, forcing me to reach out to others to publish my work. Having only taken a few college level math classes and not knowing TeX, I was unprepared for how much work it would take to write a good paper. My first problem was that I was over-reaching in both the breath and depth of what I was trying to communicate. So my main advice is that small is beautiful. There are mathematicians of significance who's dissertations had a strong impact on mathematics, but if you engage in "empire building" as a novice, people are likely not to take you seriously. 
In general no one is going to know who you are. As my knowledge of mathematics and my writing ability have grown I can now submit articles for publication and the publishers now refer to me as Doctor unless I correct them. So there is no barrier to a non-professional publishing except being unable to write at a professional level. But it took me fifteen years to get to this place by myself.
Know what you want and what you are willing to pay for it. I ended up going back to school in mathematics just so I could understand math better and I'm glad I did.
Enjoy the journey. It is now twenty five years later and I am still unpublished, but that is ok because I'm enjoying the journey. 

Note: I recommend when publishing in research journals that not only does your research need to be solid, but it should discuss the relevance of other important research in the field of your publication. You are not just adding another block of knowledge, you cementing it into place in the edifice of mathematics. Educational journals might be more accepting of stand alone material. 
A: It certainly does happen that amateur mathematicians publish articles in good journals.  Unfortunately the good amateur mathematicians have the burden of distinguishing themselves from the crackpots.
Without more information it is hard to give advice.  Everybody's ideas of a good paper and a top journal are different.  But I can say two simple things for the general case.
First, it is definitely worth the effort to make sure your writing style is in line with that of good math papers.  If your paper is written in a way that doesn't conform to writing standards/norms in the area, the referees will have a bad first impression.  Second, make sure you acknowledge anyone who helped you with the writing.  In my experience this is best done briefly in a final section at the very end of the paper.
