Erdos 1949 number terms square polynomial

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conjecture_p65: Rényi's conjecture, which Erdős records from oral communication, that the fewest terms Q(k) of the square of a real polynomial with k nonvanishing terms tends to infinity; the paper says neither could yet prove it.

remark_p65: Erdős's closing remark that, since Rényi proves Q(29) <= 28 for polynomials with rational coefficients, the proof of the Theorem gives Q(k) <= c_2 k^{1-c_1} for polynomials with rational coefficients.

theorem_p63: Erdős's theorem that there are constants c_2 > 0 and 0 < c_1 < 1 such that the fewest terms Q(k) of the square of a real polynomial with k nonvanishing terms satisfies Q(k) < c_2 k^{1-c_1}, so Q(k)/k tends to 0.

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P. Erdős: On the number of terms of the square of a polynomial, Nieuw Arch. Wiskunde (2) 23 (1949), 63--65 MR 10,354b; Zentralblatt 32,2. No copyright or license line is printed on the scan's first or last pages; the hosting archive's site footer speaks for the site, not the paper ("(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only.", https://users.renyi.hu/~p_erdos/, read 2026-10-02); the journal has no publisher page or DOI for this edition, so none was consulted, and no Crossref license is recorded; the term is unstated.

Writing Q(k) for the minimum number of terms of f(x)^2 over polynomials f with exactly k nonvanishing terms and real coefficients, Erdős proves the Theorem (p. 63) that there are constants 0 < c_2 and 0 < c_1 < 1 with Q(k) < c_2 k^{1-c_1}, which proves Rényi's conjecture that Q(k)/k tends to 0. The paper recalls that Rényi, Kalmár and Rédei had shown that the lower limit of Q(k)/k is 0 and that Rényi had shown that the averages of Q(k)/k tend to 0. The proof (pp. 64--65) modifies Rényi's method and uses two facts the paper takes from Rényi's paper without proof, Lemma I, Q(29) <= 28, and Lemma II, Q(a.b) <= Q(a).Q(b), which give Q(29^l) <= 28^l; for other k it multiplies a polynomial with a sparse square by a polynomial whose coefficients are fixed by solving linear equations. The closing paragraph (p. 65) says it would be interesting to determine the order of Q(k) more accurately, records Rényi's conjecture (oral communication) that Q(k) tends to infinity, which neither could yet prove, and remarks that since Rényi proves Q(29) <= 28 for polynomials with rational coefficients, the proof gives Q(k) <= c_2 k^{1-c_1} for rational coefficients too; it adds Rényi's question whether Q(k) is the same for rational, real or complex coefficients.

Source: https://users.renyi.hu/~p_erdos/1949-08.pdf.

Read status: claims checked. Every statement on the result pages was read clause by clause on the print, and the proof of the Theorem was read; Lemmas I and II, which the paper takes from Rényi's 1947 paper, were not checked. Result pages: theorem_p63, remark_p65 and conjecture_p65.

Bears on.

• #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 the same question for real coefficients as Rényi's conjecture (p. 65) and does not prove it; its Theorem (p. 63) and closing remark (p. 65) give the upper bound c_2 k^{1-c_1} for the real and the rational minimum.

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