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Schinzel and Zannier [ScZa09] sharpen Schinzel's theorem of 1987 (Schinzel 1987) by one logarithm. Their Theorem 1 states that for a field $k$, a polynomial $f\in k[x]$ with $T\ge2$ terms whose power $f^l$ has $t$ terms, and either $\operatorname{char}k=0$ or $\operatorname{char}k>l\deg f$,

$$t \ge 2+\frac{\log(T-1)}{\log 4l}.$$

For $l=2$ and rational coefficients this gives $f(k)\ge2+\log(k-1)/\log8$, so $f(k)\gg\log k$ and $f(k)\to\infty$, which answers Problem 485 yes with a stronger bound. The proof follows Schinzel's approach but inducts on degrees rather than on $t$, through a two-variable form built from simultaneous rational approximations to the exponent ratios; the source card digests the paper, whose Theorem 2 treats positive characteristic. The authors remark that even for $l=2$ the bound is far from the best known upper bound, Verdenius's $t\ll T^{\log8/\log13}$ along a sequence of polynomials with $T\to\infty$. The paper was received on 2008-08-28 and communicated on 2008-11-14, as its last page records, and published on 2009-03-31.

Depends on. Nothing in this wiki; the result rests on the refereed paper linked above.

Acceptance. Refereed: the paper appeared in Atti della Accademia Nazionale dei Lincei, Rendiconti Lincei Matematica e Applicazioni 20 (2009), no. 1, 95–98. Reviewed: the site's curator, Thomas F. Bloom, records the improvement $f(k)\gg\log k$ in the problem's commentary (page last edited 2026-04-08, read 2026-10-07). No formal proof of this bound is held or audited here, so no formalized evidence is listed.