The generative AI system itself cannot own copyright as only human beings can do so. As the user of the generative AI system that has used your own work and prompts to create a new work, you are potentially the copyright holder of the new work but with many provisos. Your new work has to be sufficiently original in its expression to qualify as a new work protected by copyright that you, and you alone as an individual creator, could have produced. Furthermore, if the generative AI system you have used has been trained using multiple copyright works – whether with or without permission – there is the possibility that much of your ‘new’ work is actually borrowed verbatim from these existing works.
When you generate work using generative AI it is good practice to declare what you have used and how.