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908 of 1221 statements are formalized in Lean (Formal Conjectures) · 107 carry an Erdős prize · 3 are independent of ZFC and 9 more can't be proved, or can't be disproved, from ZFC alone · “open” counts the computable and Lean-checked ones too.
If \(p(z) \in \mathbb{C}[z]\) is a monic polynomial of degree \(n\), is the length of the curve \(\{z \in \mathbb{C} : \lvert p(z)\rvert = 1\}\) maximised when \(p(z) = z^n - 1\)? We require \(n \geq 1\). For \(n = 0\) the only monic…
If \(p(z)\) is a polynomial of degree \(n\) such that \(\{z : \lvert p(z)\rvert\leq 1\}\) is connected then is it true that \[\max_{\substack{z\in\mathbb{C}\\ \lvert p(z)\rvert\leq 1}} \lvert p'(z)\rvert \leq (\tfrac{1}{2}+o(1))n^2?\]…
Let \(p(z)=\prod_{i=1}^n (z-z_i)\) for \(\lvert z_i\rvert \leq 1\). Is it true that \[\lvert\{ z: \lvert p(z)\rvert <1\}\rvert>n^{-O(1)}\] (or perhaps even \(>(\log n)^{-O(1)}\))? Conjectured by Erdős, Herzog, and Piranian [EHP58]. The…
Is it true that \(\limsup M_n = \infty\)? This is Problem 4.1 in [Ha74] where it is attributed to Erdős. The weaker conjecture that \(\limsup M_n=\infty\) was proved by Wagner [Wa80], who show that there is some \(c>0\) with…
Let \(P(z)=\sum_{1\leq k\leq n}a_kz^k\) for some \(a_k\in \mathbb{C}\) with \(\lvert a_k\rvert=1\) for \(1\leq k\leq n\). Does there exist a constant \(c>0\) such that, for \(n\geq 2\), we have \[\max_{\lvert z\rvert=1}\lvert P(z)\rvert…
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…
Let \(f(z) ∈ ℂ[z]\) be a monic non-constant polynomial. Can the set \(\{z ∈ ℂ : |f(z)| ≤ 1\}\) be covered by a set of closed discs the sum of whose radii is \(≤ 2\)?
Let \((\epsilon_k)_{k\geq 0}\) be independently uniformly chosen at random from \(\{-1,1\}\). If \(R_n\) counts the number of real roots of \(f_n(z)=\sum_{0\leq k\leq n}\epsilon_k z^k\) then is it true that, almost surely, \[\lim_{n\to…
Let \(f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k\) be a random polynomial, where \(\epsilon_k\in \{-1,1\}\) independently uniformly at random for \(0\leq k\leq n\). Is it true that, if \(R_n\) is the number of roots of \(f(z)\) in…
Is it true that all except at most \(o(2^n)\) many degree \(n\) polynomials with \(\pm 1\)-valued coefficients \(f(z)\) have \(\lvert f(z)\rvert <1\) for some \(\lvert z\rvert=1\)? The answer is yes: Littlewood [Li66] conjectured the…
For an irreducible polynomial \(f \in \mathbb{Z}[x]\) with \(f(n) \ge 1\) for sufficiently large \(n\), does there exists a constant \(c = c(f) > 0\) such that \(\sum_{n \le x} \tau(f(n)) \approx c \cdot x \log x\)? Note that it is unclear…
**Erdős–Herzog–Piranian Component Lemma** (Metric Properties of Polynomials, 1958): If \(f\) is a monic degree \(n\) polynomial with all roots in the unit disk, then some connected component of \(\{z \mid |f(z)| < 1\}\) contains at least…
Let \(C>0\). There exists \(\epsilon>0\) such that if \(n\) is sufficiently large the following holds. For any \(x_1,\ldots,x_n\in [-1,1]\) there exist \(y_1,\ldots,y_n\in [-1,1]\) such that, if \(P\) is a polynomial of degree…
Is there some constant \(c > 0\) such that, for all large enough \(n\) and all polynomials \(P\) of degree \(n\) with coefficients in \(\{-1, 1\}\), \[\max_{|z|=1} |P(z)| > (1 + c) \sqrt{n}?\] No. This was disproved by an internal OpenAI…