Problem 485

Let $f(k)$ be the minimum number of terms in $P(x)^2$ , where $P\in \mathbb{Q}[x]$ ranges over all polynomials with exactly $k$ non-zero terms. Is it true that $f(k)\to\infty$ as $k\to \infty$ ?

Source: erdosproblems.com/485

Analysis
Polynomials

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SolvedProved

An accepted solution exists. The statement is true.

Proved. The site labels the problem PROVED. The answer is yes: Schinzel proved in 1987 that $f(k)>\log\log k/\log2$ , and Schinzel and Zannier sharpened this in 2009 to $f(k)\gg\log k$ (both refereed; the accepted claim pages are Schinzel 1987 and Schinzel and Zannier 2009).

Er49b: Erdős, P., On the number of terms of the square of a polynomial. Nieuw Arch. Wiskunde (2) (1949), 63-65.
Ha74: Hayman, W. K., Research problems in function theory: new problems. (1974), 155-180.
Re47: Rényi, A., On the minimal number of terms of the square of a polynomial. Hungarica Acta Math. (1947), 30-34.
Sc87: Schinzel, A., On the number of terms of a power of a polynomial. Acta Arith. (1987), 55-70.
ScZa09: Schinzel, Andrzej and Zannier, Umberto, 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.

Problem 485

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