Powerful values of polynomials and a conjecture of Vojta

We study some problems about powerful values of polynomials over number fields, such as giving uniform bounds for the number of consecutive squareful values of squarefree polynomials, or the higher exponent analogue of the M squares problem. We show that a Diophantine conjecture of Vojta implies complete answers to these problems, and we show unconditional analogues for function fields and complex meromorphic functions. Some of these results have consequences in logic related to Hilbertʼs tenth problem, and we also explore these.