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Source. 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

Here $Q(k)$ is as on the Theorem page: the minimum number of terms of $f_k(x)^2$ over polynomials $f_k$ with $k$ nonvanishing terms and real coefficients.

Conjecture (p. 65, attributed to Rényi, oral communication). $\lim_{k\to\infty}Q(k)=\infty$.

The paper says it would be interesting to determine the order of $Q(k)$ more accurately, and that neither Rényi nor Erdős could yet prove the conjecture.

Read depth. Claims checked: the paragraph was read on the print.

Proof pointer

None; the paper records the conjecture and gives no argument for it.

Dependencies

None.

Bears on

• Problem 485: the problem asks whether the fewest terms of the square of a rational polynomial with exactly $k$ nonzero terms tends to infinity. The paper records this conjecture for real coefficients, and Rényi's question whether $Q(k)$ depends on the coefficient field, and proves neither.