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Source. Closing remark, p. 65, of P. Erdős, "On the number of terms of the square of a polynomial," Nieuw Arch. Wiskunde (2) 23 (1949), 63--65. The edition read is identified on the source card.

Statement

Remark (p. 65, unnumbered). Since Rényi proves $Q(29)\le28$ for polynomials with rational coefficients, the proof of the Theorem gives $Q(k)\le c_2k^{1-c_1}$ for polynomials with rational coefficients: the minimum number of terms of $f_k(x)^2$ over polynomials $f_k$ with $k$ nonvanishing terms and rational coefficients is at most $c_2k^{1-c_1}$.

The paper adds that Rényi asks whether $Q(k)$ is the same when the coefficients are rational, real or complex; it does not answer this.

Read depth. Claims checked: the remark was read clause by clause on the print. The paper gives no separate argument, and that every step of the Theorem's proof stays within rational coefficients was not checked here.

Proof pointer

The paper's reason is the one stated: Rényi's example for $Q(29)\le28$ has rational coefficients, and the proof of the Theorem builds its polynomials from that example by multiplication and by solving linear equations of the first degree.

Dependencies

Theorem (p. 63) and its proof; Rényi's rational example for $Q(29)\le28$.

Bears on

• Problem 485: the problem's $f(k)$ is the rational minimum this remark bounds, so the remark gives the upper bound $f(k)\le c_2k^{1-c_1}$; it does not decide whether $f(k)\to\infty$.