Problem 485

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claims/: The 2 claim pages of Problem 485, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $k\to \infty$?

Status. 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).

Source. erdosproblems.com/485, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #485, https://www.erdosproblems.com/485.

References.

• [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.

Formalization. Statement in formal-conjectures, pinned to the commit that last changed it, which leaves its theorem unproved and points to a Lean 4 proof in the lean-proofs repository; the claim page for Schinzel 1987 links that proof at its pinned commit.

Current assessment

The question (site formulation, 2026-09-04). Whether the least number $f(k)$ of nonzero terms in the square of a rational polynomial with exactly $k$ nonzero terms tends to infinity with $k$. PROVED. The problem is Problem 4.4 of Hayman's collection [Ha74], attributed to Erdős; the site's commentary (page last edited 2026-04-08) records that Rényi and Rédei first studied the question, that Erdős [Er49b] proved the upper bound $f(k)<k^{1-c}$ for some $c>0$, that the conjecture $f(k)\to\infty$ is due to Erdős and Rényi, and that Schinzel [Sc87] solved it.

Standing. Two accepted full claims, both refereed: Schinzel 1987, which proves $f(k)>\log\log k/\log2$ as the square case of a lower bound for the terms of any power of a polynomial over a field of characteristic zero or large characteristic, and Schinzel and Zannier 2009, which sharpens it to $f(k)\ge2+\log(k-1)/\log8$. Both hold over every field of characteristic zero, $\mathbb{C}$ and $\mathbb{Q}$ included, and over fields of large characteristic, so they bound the rational minimum from below. The problem's standing follows from them.

Bounds. Erdős's upper bound (card) is $Q(k)<c_2k^{1-c_1}$ for the real minimum, obtained from Rényi's example $Q(29)\le28$ and submultiplicativity; since Rényi's example has rational coefficients, the paper's closing remark (p. 65) gives $f(k)\le c_2k^{1-c_1}$ as well. Schinzel and Zannier quote Verdenius's sharper upper bound $t\ll T^{\log8/\log13}$ along a sequence of polynomials, and a comment of 2026-02-01 on the site's thread records Verdenius's exponents for real and for integer polynomials as the best known. The gap between $\log k$ and a power of $k$ is open and is not part of the problem's question.

Formalization. The site's label carries no Lean qualification. The problem's statement is formalized in formal-conjectures, whose file leaves its theorem unproved and points to a Lean 4 file in the lean-proofs repository that declares itself a formalization of Schinzel's solution and proves that the minimum tends to infinity; the claim page links it at its pinned commit. Nothing was built or audited here, and no formalized evidence is claimed.

Search scope (2026-10-07). The site's page and thread, the formal-conjectures file and the lean-proofs file; statements and dates from the papers (Erdős 1949, Theorem; Schinzel 1987, Theorem 1; Schinzel and Zannier 2009, Theorem 1).

Progress

Schinzel's theorem [Sc87] settles the question, and Schinzel and Zannier [ScZa09] improve the lower bound by one logarithm; Erdős [Er49b] supplies the power-saving upper bound.

Known Results

• [Sc87], Schinzel, Theorem 1: for $f$ with $T\ge2$ terms over a field of characteristic zero or greater than $l\deg f$, $f^l$ has at least $l+1+(\log2)^{-1}\log(1+\log(T-1)/(l\log4l-\log l))$ terms; for $l=2$, $f(k)>\log\log k/\log2$.
• [ScZa09], Schinzel and Zannier, Theorem 1: under the same hypotheses $f^l$ has at least $2+\log(T-1)/\log4l$ terms; for $l=2$, $f(k)\gg\log k$.
• [Er49b], Erdős, Theorem: the real minimum satisfies $Q(k)<c_2k^{1-c_1}$ for constants $c_2>0$ and $0<c_1<1$, so $Q(k)/k\to0$, confirming Rényi's conjecture; the closing remark (p. 65) gives $f(k)\le c_2k^{1-c_1}$ as well.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1949_number_terms_square_polynomial
• erdos_1949_number_terms_square_polynomial / conjecture_p65
• erdos_1949_number_terms_square_polynomial / remark_p65
• erdos_1949_number_terms_square_polynomial / theorem_p63
• schinzel_1987_number_terms_power_polynomial
• schinzel_2009_number_terms_power_polynomial
• schinzel_2009_number_terms_power_polynomial / theorem_1