There is no hard and fast rule that says your Ph.D. advisor must be listed as an author on your work. It may be unethical to include the name of someone who had no hand in creating or generating the paper or the ideas when you provide anything fresh that stems from your thesis. Even though your Ph.D. advisor is listed as an author on a previous paper, they need not appear as such on a new paper. Your actions may cost you politically with your supervisor, but if the study was yours and not his/hers, you have a moral case. It is preferable to inform them now and risk their anger than to let them find out later.
There are two possible outcomes here. Depending on your study area, one possibility is not so dire, while the other might be disastrous to your academic future.
The first situation is appropriate if the piece is a post-Ph.D. Review the same papers included in your thesis’s required reading list. Of course, it’s good if you want to write something together, but it’s not needed. Therefore, you should tell them. In the second situation, if it is directly related to your Ph.D. thesis and you are working in the medical, life sciences, or many physical disciplines, it was not a smart move.
Because of this, there is no need to tell them about it. There is no ulterior motive for publishing. Being honest about it and trying to mitigate its harmful impacts is still in your best interest, even though it will be publicly known. Not that this will ever result in anything good, but it beats the alternative.
Publishing joint work with your supervisor without crediting them as an author is likely academic misconduct unless you get prior approval from a higher-up in the department, such as the chair or dean. If you choose to tackle this problem alone, you have just set off a nuclear bomb in a mentor-mentee relationship. There’s no avoiding the truth and no turning back now.
Your supervisor has the right (and is likely to exercise it) to disavow the work you submitted to the publication because it was done under their supervision but was not listed as a co-author. Your paper will be withdrawn as a consequence. Since the grounds for this kind of retraction are usually made public, it is one of the worst that may happen.
If an unnamed collaborator ruins your work, there’s almost no possibility of submitting it somewhere else. If you are still enrolled in a Ph.D. program, your advisor may petition the department to have you dismissed for academic dishonesty. That would be well within their legal rights. If the latter is the case, the takeaway should be to avoid making unilateral power movements while in a weak position.
To sum up, the answer to the question of significance is that it is conditional. It may have no effect or lead to allegations of academic misconduct that would severely damage your reputation and job prospects. Ultimately, you have to ask yourself, “Did you do anything wrong?”. If your paper contains someone else’s data or represents someone else’s work, if you failed to include one or more people who should have been listed as co-authors, if you knew your advisor would disapprove but submitted it anyway, or if you did any of several other wrong things, you will get in trouble.
Your advisor’s petty and vengeful nature will also bring difficulties to you. It may not be that horrible if none of these things happen. It would be best if you visited your supervisorâ€™s office, informed them of the publishing, and explained why you failed to inform them beforehand.