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a) Does anyone know of any interesting theory related to them that I could read up upon?

b) How would one start solving them?

Here are the problems:

1) Show that there are infinitely many polynomials p with integer coefficients such that P(x^2 - 1) = (P(x))^2 - 1, P(0) =0 .

2) Are there real polynomials p satisfying P(x^2 - 1) = (P(x))^2 + 1 for all x? If so, determine what they look like.

Observe the plus sign at the end of the second problem.