So you advisor gave you an idea and you want to publish a solo article based on that idea?
Okay, I am not a mathematician and not Italian. Every research field is different, and every country is different. Maybe this is totally fine for mathematics in Italy. Maybe your professor will say, "Yeah, sure, go for it!"
But that would definitely not go in Japan, at least in most research fields. Everyone would be appalled by the fact that this came to your mind. A solo article is when you come up with an idea yourself, do all the work yourself, and write the manuscript yourself. And if you don't want to find yourself in the grey area of ethics, you shouldn't get any help or advice whatsoever in relation to your article. Ideally, you don't even let anyone know until the paper is published - otherwise people may give you advice and you won't be able to unhear it. And it's really not nice if you publish a solo article while being funded from a grant secured by someone else.
(Update: a comment below says that in mathematics, merely asking a question without contributing to its solution does not normally justify authorship, and obtaining funding doesn't either. Lucky you mathematicians. But read on. There's a totally different thing to consider.)
Another aspect is that even if you come up with an idea yourself, do all the work yourself, write the manuscript yourself, are being paid not from anyone's grant, and are thus perfectly positioned from the ethics standpoint to submit a solo article, you might later find yourself truly regretting publishing it. It happened to one of my colleagues. He wasn't experienced enough. He wishes he could unpublish it.
Look, you are a PhD student and don't have a lot of experience. Are you really sure you can write a paper you won't later regret publishing? Are you sure you won't overlook a serious issue?
There are a multitude of ways to screw up. If you make an error or a wrong statement in your paper, someone may publish a comment, and it will be a huge blow. If you write your paper poorly in terms of the language, structure, clarity, etc., this will seriously affect how others see you. If you publish your paper in a subpar journal, others will think it's what you and your work is worth. Your first papers set the expectations of others - it's called the anchoring effect. if you submit your paper to a high-profile journal and the referees say that your work falls very short of meeting the acceptance criteria, the editor may well have prejudice against you next time - that is, when you submit another paper to that journal. Remember, the editor chooses the referees, so you don't want the editor to have prejudice against you. And this might make it almost impossible for you to publish anything in that journal later. Also, if you publish an article and no one cites it, it won't look good to anyone looking at the citation stats of your papers - and that's what many referees do.
If you have your professor as a co-author, he'll ensure your paper won't have any serious issues. It's about his or her reputation. And having your professor as a co-author is likely to have a positive effect on the recognition of the paper by the referees and readers as well as on its citations.
My advice is: think twice whether you really want to proceed with this on your own and publish a solo paper. That's the very first thing to decide on. Make sure to take a well-thought decision. If you do decide to talk with your professor, do your best and utmost to ensure he or she won't feel bad about you. One of the key factors is whether you are a paid a salary or a stipend from a grant secured by your professor. If that's the case, he might feel as if you were trying to steal his money. Remember, you depend on your professor. The relationship is extremely important. Focus on that.
Hey boss, I've noticed some of your PhD students published solo
papers. Is there any chance I can do the same and you'll be fine with
it? If so, how?
That's a good start, and his reaction will show whether it's a good idea to talk about that particular paper.