If you are a full time researcher, you may not have the luxury of an adviser to guide you, especially if it's a new field. But, as a graduate student, you should seriously discuss with your adviser what you have read and what you found interesting. They should be able to help you find that research topic.
Ask them to send you to a summer school or conference, where you would get the pulse of what's going on in your subject area. That's also the place to meet other people who might become later your friends and collaborators.
Apart from the adviser, you should talk to the people in your thesis committee, if you have one. If your department is a healthy place, they would be more than happy to help. Also, talk to other faculty and students about things related to the field you want to launch yourself into.
It is not sufficient to read a lot in order to come up with ideas of research subjects. From reading you may find out what are the main open problems in your field and what are the state-of-the-art methods. But, there is no substitute for hands on research experience. Without that, you cannot know if the thesis topic you propose is something doable or not.
Since you read a lot about computational algebra, try to find a review where they talk about open problems in the field. Pick one which sounds interesting and do some more literature search to see what you need to know to approach the subject. Then discuss that with your adviser, peers or other faculty that might be helpful. They should help you divide this problem into subproblems, and one of them might be your topic.