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## Main Question or Discussion Point

Here are two very similar questions about polynomials that I feel may have deeper roots (excuse pun).

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.

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.