FormalConjectures.ErdosProblems.«485»

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module

public import FormalConjecturesUtil

Erdős Problem 485

References:

erdosproblems.com/485

[Re47] Rényi, A., On the minimal number of terms of the square of a polynomial. Hungarica Acta Math. (1947), 30-34.

[Er49b] Erdős, P., On the number of terms of the square of a polynomial. Nieuw Arch. Wiskunde (2) (1949), 63-65.

[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.

[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.

[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. (2009), 95-98.

@[expose] public sectionopen Filter Polynomialnamespace Erdos485

The minimum number of terms of the square of a rational polynomial with exactly k nonzero terms, where the number of terms of P is P.support.card.

noncomputable def f (k : ℕ) : ℕ :=
  sInf {m | ∃ P : ℚ[X], P.support.card = k ∧ (P ^ 2).support.card = m}

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 $k \to \infty$?

A conjecture of Erdős and Rényi (this is Problem 4.4 in [Ha74], attributed to Erdős); the function was first investigated by Rényi and Rédei [Re47], and Erdős [Er49b] proved that $f(k) < k^{1-c}$ for some $c > 0$. The answer is yes: Schinzel [Sc87] proved $f(k) > \log \log k / \log 2$, and Schinzel and Zannier [ScZa09] improved this to $f(k) \gg \log k$.

@[category research solved, AMS 11 12, formal_proof using lean4 at
  "https://github.com/plby/lean-proofs/blob/8822f7ddef30fadbd92e1c6ab4ed897af356af5e/src/latest/ErdosProblems/Erdos485.lean#L39"]
theorem erdos_485 : answer(True) ↔ Tendsto f atTop atTop := by⊢ True ↔ Tendsto f atTop atTop
  sorryAll goals completed! 🐙

end Erdos485