Polynomials: Theory and Problems to Solve

[jonas.hall]

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.

 

[fresh_42]

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?

Algebraic geometry.

b) How would one start solving them?

Solve what? If you are looking for roots, then a numerical algorithm is probably the best way to approach the problem. Unless there are additional information on the number of variables, coefficients and degrees.

Here are the problems:

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

Make an ansatz $P(y)=a_ny^n + \ldots +a_1y+a_0$ and see what these conditions mean to your coefficients.

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.

Same idea as above. Assume an arbitrary solution and deduce conditions for the coefficients. If you run into a contradiction, then there won't be a solution. Otherwise the conditions will tell you how to chose $P$.