Most applicants to a mid- to upper-tier math Ph.D. program will be coming directly from an undergraduate program, and the core planks of their application will be their grades, letters of reference from professors, and (in many cases) good performance on mathematics competitions. Some but very few will have published papers in peer-reviewed journals. All of these will be viewed as predictors of future, to be developed, mathematical research ability.
Your application will be unusual in many ways, your work experience, the absence of some of the typical predictors (grades, etc), and this paper (congratulations, by the way!).
To the extent a committee singles out your application for discussion, your paper should go a long way to reassuring concerns you may be a risky admit. Someone should say, "the grades are poor, but they clearly demonstrably can and want to do math research. Why are we trying to predict the potential of future research ability when we have evidence of it already being in place." However, you still run 2 risks:
Your application may be rejected out of hand due to the usual metrics, esp. grades, field of study, etc. and not get serious thoughtful consideration at all
The committee may be risk averse, and if it can fill its spots with "typical" candidates, they may do so, especially if they have recommendations from math professors they know, good performance on the Putnam competition, etc.
As to your publication being sole-authored, as others have written, in math this is not at all unusual. Sole- or 2-authored papers are the norm. In your case, there is still benefit that it demonstrates you did the work on your own, rather than being a hanger-on in a team effort. However, given the unusual situation, frankly you would be equally well-positioned with a co-authored paper with a research mathematician with that co-author writing you a letter of reference describing how you were the lead author.
In any case, you have something good, so congratulations. And you know what you want, so go for it. Given that, a few suggestions to put your best foot forward:
Is there any professional mathematician you have talked to about your paper, who helped you with something, who you've collaborated on about something else? It would be very helpful for you to have a letter of reference with discusses and puts in context your research accomplishments, in addition to just the paper itself.
Broadly speaking, when I am on a committee, I know direct-from-undergrad applicants, no matter how excellent, are a blank slate. So I expect them to have not much idea exactly what they want to do, to not know what they don't know, so to speak. The more an applicant is mature/from industry/track change, the more I expect them to have a plan, a research idea, to have done background reading, etc. All of these are indicia they have thought through what they want, that they have learned skills from their life experience so far, and that they are pivoting with forethought rather than to get out of a dead-end - and might give up and pivot away from us with minimal provocation. With your paper to open the door, more so than others I would think you have a lot to gain from identifying specific individuals at specific universities you'd like to work with, and reaching out to them directly (with your paper) prior or in parallel to the normal application process
Good luck!
(My background - Ph.D. in pure mathematics in the US, then pivoted to industry, now straddling academia and industry in a multi-disciplinary field)
