The analysis proves quartic polynomials cannot produce all non-negative integers, indicating limitations in mathematical representation.
We prove that there does not exist F ∈ Q[x,y] of degree 4 such that F(Z²) = Z≥ 0 . In particular, this answers a question by John S. Lew and Bjorn Poonen for quartic polynomials.
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Xiao et al. (2025) studied this question.
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