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I am applying for pure Math Ph.D programs this year.

I did some independent research in centralizer of matrices, and I developed an algorithm to generate the explicit k-basis for the space of centralizers. I guess this problem is not solved since I cannot find any papers on this topic, and a similar question asked on Math Stack Overflow has no complete solutions.

I asked my professor (one of my recommendation letter writer) to take a look, but this is irrelevant to his fields of studies and he said that he might have no time to read this.

I want this research experience to strengthen my application. So, how should I deal with my paper? Shall I just mention it in my resume and personal statement or should I post the paper on arXiv? I am not sure what is appropriate.

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    How about you post the paper on arXiv and then link it as the answer to the question on Math Overflow? This way you can hopefully get some feedback on your work. You can decide what to do next based on the response you get.
    – Petr Naryshkin
    Commented Sep 28, 2019 at 0:27
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    I disagree with @PetrNaryshkin's advice. I would not post the paper to arXiv until after you have been able to have someone look at. (If you already have submitted, you should be able to unsubmit within 24 hours or so.) arXiv preprints can never be completely removed once posted, and people expect them to be of good professional quality. I don't know you or your work, but it's often hard for undergraduates working alone to produce work up to that standard, or to even be able to tell whether they have achieved it. ...
    – Nate Eldredge
    Commented Sep 28, 2019 at 1:33
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    If the quality is not what people expect, it may be harmful to your application or to your future career. Also, MathOverflow is not really a good way to get general feedback on your work; see meta.mathoverflow.net/questions/927/… for instance.
    – Nate Eldredge
    Commented Sep 28, 2019 at 1:35
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    @Nate Eldredge Thanks for your advice. I didn't submit the paper when I saw the the no complete removal policy. I got rejections from all Ph.D programs that I applied two years ago when I applied as an undergraduate (Partially because I only applied top 30s). I am second year in master program now, and I really want to have a strong profile for applications.
    – TH Wang
    Commented Sep 28, 2019 at 5:59
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    I do know that my paper only used some standard grad-level Algebra, and the main theorem deduced is trivial in many senses. I am still proud of what I did since this is the first time that I proved some original results on my own. I really want to include this experience in my application, to show that I have research potentials besides of just doing well in standard courses. I got confused about what is appropriate to do.
    – TH Wang
    Commented Sep 28, 2019 at 5:59

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If your goal is to have a way to include this in your graduate applications, you can certainly talk about this project in a statement of interest. Search this site for "statement of interest" and you will see many have no idea what to put there. You have something concrete to talk about! Did this experience make you want to study more algebra, or are you ready to switch to analysis? Do you want to look at applications, or more theory, of matrices over finite fields?

It is a slight disappointment that you did not do this as an independent study, which is what someone my age would have done. On the other hand, you did get feedback on MathOverflow. Graduate admissions committees can take into account interactions like this. I would say this shows independence and a willingness to learn by whatever means available. You cannot say that in a statement of interest, but you can talk about how you enjoy having Denis Serre answer a question of yours. (Name-dropping is best done shamelessly.)

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