What equations (polynomials involving one variable) result in the fundamentals?

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n=1; x^2 (perhaps with some shift that elongates the parabola

n=2; x^3 (This should work as well)

I think it continues like this with some extra terms that are less than the highest degree of the equation.

This brought up the investigation of these polynomials, this have some interesting properties/shape.

y=x^3-x^2-x
y=x^4-x^3-x^2-x
y=x^5-x^4-x^3-x^2-x