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Statement

Theorem 2 (p. 96). Let $\operatorname{char}k>0$, $f\in k[x]$ and $l\in\mathbb N$, and suppose $f$ has $T\ge2$ terms and $f^l$ has $t$ terms. If

$$l^{t-1}(T^2-T+2)<\operatorname{char}k,$$

then

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

The paper calls this a supplementary result in positive characteristic (p. 95). Unlike Theorem 1, its hypothesis bounds the characteristic from below in terms of $l$, $t$ and $T$ rather than $l\deg f$.

Proof pointer

P. 98. The paper states only that Theorem 2 follows from Theorem 1 in the same way as Theorem 2 follows from Theorem 1 in Schinzel's 1987 paper; no further argument is printed.

Read depth

Claims checked: the statement and hypotheses were read clause by clause on the print. The deduction is referred to Schinzel 1987 and was not read. Nothing here is independently reviewed.

Dependencies

Theorem 1 of the paper, and the deduction of Theorem 2 from Theorem 1 in A. Schinzel, On the number of terms of a power of a polynomial, Acta Arith. 49 (1987), 55--70.

Source. A. Schinzel and U. Zannier, On the number of terms of a power of a polynomial, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 20 (2009), no. 1, 95--98, doi:10.4171/RLM/534; the edition read is named on the source card.

Bears on

None directly: Problem 485 concerns rational polynomials, which Theorem 1 covers.