---

Statement

Setting (p. 95). The number of terms of a polynomial is the number of its non-zero coefficients.

Theorem 1 (p. 95). Let $k$ be a field, $f\in k[x]$ and $l\in\mathbb N$, and suppose $f$ has $T\ge2$ terms and $f^l$ has $t$ terms. If $\operatorname{char}k=0$ or $\operatorname{char}k>l\deg f$, then

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

The paper presents this as removing, roughly, one logarithm from Schinzel's 1987 bound, which in characteristic zero had the shape $t\ge c_l\log\log T$ with an explicit $c_l>0$ (p. 95). It adds (p. 96) that even for $l=2$ the bound is far from the best known upper bound, due to Verdenius: $t\ll T^{\log8/\log13}$ for a sequence of polynomials whose number of terms tends to infinity.

Proof pointer

Pp. 97--98. The proof takes $f_0$ of least degree violating (1), so that $T>1+(4l)^{t-2}$ (inequality (3), p. 97), and applies Lemma 2 (pp. 96--97, quoted from Schinzel's 1987 paper, proof there on pp. 60--63) with exponent ratios approximated by Dirichlet's theorem. This yields a two-variable identity $F=cF_0^l$; substituting $y=x^q$, $z=x$ for a suitable $q$ and using Lemma 1 (p. 96, Schinzel 1987, Lemma 2) produces $f_1=cg_1^l$, where $f_1$ has at most $t$ terms and $g_1$ has at least $T$ terms. The minimality of the degree $n_{t-1}$ of $f_0^l$, set against an upper bound for $\deg f_1$, then gives $T<1+(4l)^{t-2}$, contradicting (3). The authors name the new ingredient relative to Schinzel 1987 as an induction on degrees rather than on $t$ (p. 96).

Read depth

Claims checked: the statement, its hypotheses and the comparison with Verdenius were read clause by clause on the print, and the proof on pp. 97--98 was followed. The two lemmas are quoted from Schinzel 1987 and their proofs were not read. Nothing here is independently reviewed.

Dependencies

None in the corpus. External inputs: Lemmas 1 and 2 of the paper, both taken from A. Schinzel, On the number of terms of a power of a polynomial, Acta Arith. 49 (1987), 55--70; Dirichlet's approximation theorem.

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

• Problem 485: the problem asks whether the least number $f(k)$ of terms of $P(x)^2$, over $P\in\mathbb Q[x]$ with exactly $k$ non-zero terms, tends to infinity. Theorem 1 with $l=2$ and a field of characteristic zero gives $f(k)\ge2+\log(k-1)/\log8$ for every $k\ge2$, so $f(k)\to\infty$ and indeed $f(k)\gg\log k$.