Squares of sparse polynomials

DefinitionErdős–Rényi conjecture on squares of sparse polynomials#✓fully proved

There is a function \(f : \mathbb{N} \to \mathbb{N}\) such that for every integer \(k \ge 1\), every complex polynomial \(p\) with at least \(f(k)\) nonzero coefficients has a square \(p^2\) with at least \(k\) nonzero coefficients.

def conjecture : Prop :=
  ∃ f : ℕ → ℕ,
    ∀ k : ℕ, 0 < k →
      ∀ p : Polynomial ℂ,
        f k ≤ p.support.card → k ≤ (p ^ 2).support.card

Used by 1

Used bySchinzel's theorem on squares of sparse polynomials

TheoremSchinzel's theorem on squares of sparse polynomials#●ready to prove

The Erdős–Rényi conjecture holds: there is a function \(f : \mathbb{N} \to \mathbb{N}\) such that for every integer \(k \ge 1\), every complex polynomial \(p\) with at least \(f(k)\) nonzero coefficients has a square \(p^2\) with at least \(k\) nonzero coefficients.

Schinzel (Acta Arith. 49 (1987) 55-70) proved that the number of terms of \(g \in \mathbb{C}[x]\) is bounded in terms of the number of terms of \(g^2\); Fuchs, Mantova and Zannier (arXiv:1412.4548, Thm 1.1) give a uniform version.

public theorem_wanted conjecture_holds : conjecture

Sources

• C. Fuchs, V. Mantova and U. Zannier, On fewnomials, integral points and a toric version of Bertini's theorem (2017), arXiv:1412.4548, https://arxiv.org/abs/1412.4548.

Uses 1

Statement usesErdős–Rényi conjecture on squares of sparse polynomials

1 fully proved1 ready to prove

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