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Brando
Brando
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Suppose $f$ and $g$ are polynomials with integral coefficients and $f(\mathbb Z)\subset g(\mathbb Z)$. Is there any relation between $f$ and $g$?

For instance, this happens if $f=g\circ h$ for some third integral polynomial $h$. In the particular case that $f$ takes on only square values, we could deduce that the converse is true: $f$ is a square, so $f=g\circ h$ where $h=(ax+b)$$h$ is some polynomial and $g(x)=x^2$. Does something like this hold more generally?

Suppose $f$ and $g$ are polynomials with integral coefficients and $f(\mathbb Z)\subset g(\mathbb Z)$. Is there any relation between $f$ and $g$?

For instance, this happens if $f=g\circ h$ for some third integral polynomial $h$. In the particular case that $f$ takes on only square values, we could deduce that the converse is true: $f$ is a square, so $f=g\circ h$ where $h=(ax+b)$ and $g(x)=x^2$. Does something like this hold more generally?

Suppose $f$ and $g$ are polynomials with integral coefficients and $f(\mathbb Z)\subset g(\mathbb Z)$. Is there any relation between $f$ and $g$?

For instance, this happens if $f=g\circ h$ for some third integral polynomial $h$. In the particular case that $f$ takes on only square values, we could deduce that the converse is true: $f$ is a square, so $f=g\circ h$ where $h$ is some polynomial and $g(x)=x^2$. Does something like this hold more generally?

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Brando
Brando
  • 671
  • 3
  • 9

When is the image of an integral polynomial contained in the image of another?

Suppose $f$ and $g$ are polynomials with integral coefficients and $f(\mathbb Z)\subset g(\mathbb Z)$. Is there any relation between $f$ and $g$?

For instance, this happens if $f=g\circ h$ for some third integral polynomial $h$. In the particular case that $f$ takes on only square values, we could deduce that the converse is true: $f$ is a square, so $f=g\circ h$ where $h=(ax+b)$ and $g(x)=x^2$. Does something like this hold more generally?