In other words, there can not be two different polynomials with integer coefficients that evaluate to the same real number when $x=e$.
2026-02-22 21:27:56.1771795676
Is it true that evaluating a polynomial with integer coefficients at $e$, uniquely defines it?
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Yes, because $e$ is not algebraic number.
See here: https://en.wikipedia.org/wiki/Transcendental_number