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Bma
Bma
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Clearly this is impossible for $p$ of even degree, and I imagine that Cardano’s formula quickly reveals it to be impossible in the cubic case, although I have not checked in detail. My guess is that no such $p$ exists. Is the answer knownDoes one exist? If one existsso, is there an explicit example?

Failing a general yes or no answer, are there sufficient conditions to identify a non-surjective polynomial function?

Clearly this is impossible for $p$ of even degree, and I imagine that Cardano’s formula quickly reveals it to be impossible in the cubic case, although I have not checked in detail. My guess is that no such $p$ exists. Is the answer known? If one exists, is there an explicit example?

Clearly this is impossible for $p$ of even degree, and I imagine that Cardano’s formula quickly reveals it to be impossible in the cubic case, although I have not checked in detail. My guess is that no such $p$ exists. Does one exist? If so, is there an explicit example?

Failing a general yes or no answer, are there sufficient conditions to identify a non-surjective polynomial function?

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Bma
Bma
  • 531
  • 3
  • 11

Does there exist some $p(x) \in \mathbb{Q}[x]$, deg$(p) > 1$, which maps $\mathbb{Q}$ onto itself surjectively?

Clearly this is impossible for $p$ of even degree, and I imagine that Cardano’s formula quickly reveals it to be impossible in the cubic case, although I have not checked in detail. My guess is that no such $p$ exists. Is the answer known? If one exists, is there an explicit example?