Polynomial Inequalities

Algebra Level 5

Suppose f(x)f(x) and g(x)g(x) are non-constant polynomials with integer coefficients, such that ff is monic and f(g(x))=(f(x))2g(x).f(g(x))=(f(x))^2\cdot g(x). Suppose NN is the number of possible polynomials g,g, such that all coefficients of gg have absolute value strictly less than 1010010^{100}. Find the last three digits of N.N.

Details and assumptions

A polynomial is monic if its leading coefficient is 1. For example, the polynomial x3+3x5 x^3 + 3x - 5 is monic but the polynomial x4+2x36 -x^4 + 2x^3 - 6 is not.

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