The criterion is that p in B must halt before returning to a Program Counter value (what they define as repeating a Turing machine "state"). That's the length o

The criterion is that p in B must halt before returning to a Program Counter value (what they define as repeating a Turing machine "state"). That's the length of the program text. So, I have no idea why they are talking about polynomials, it looks linear because the program counter has 1 value per line of code in a program text. If not, I would like to know where my misunderstanding could be. Clearly the authors think the paragraph is a simple, obvious step not worth discussing clearly (they say something about running the program for n steps being sufficient, but they do not outright, explicitly say why it would be polynomial, etc.)