Unique Monic Polynomials

Algebra Level 5

How many monic polynomials with real coefficients of degree <1000<1000 have the property that f(x2+x)=(f(x))2+f(x)f(x^2+x)=(f(x))^2+f(x) for all xx?

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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