Open problem, Analysis, posed 1958

Sendov's conjecture

Partially solved

Let $f(z) = (z - z_1)(z - z_2)\cdots(z - z_n)$ be a polynomial of degree $n \ge 2$ with complex coefficients whose zeros $z_1, \dots, z_n$ all lie in the closed unit disk $\overline{\mathbb{D}} = \{z \in \mathbb{C} : |z| \le 1\}$ . Does every closed disk $\{z \in \mathbb{C} : |z - z_k| \le 1\}$ of radius $1$ centered at a zero $z_k$ of $f$ contain at least one critical point $\zeta$ of $f$ (meaning $f'(\zeta) = 0$ )?

Research frontier as of 2026

Following Terence Tao's 2020 proof (published in 2022) that Sendov's conjecture holds for all degrees $n \ge n_0$ , only a finite range of intermediate degrees $9 \le n < n_0$ remains open in principle. However, because Tao's proof relies on compactness and contradiction arguments for the regime where the distinguished root $z_k$ is close to $0$ , the constant $n_0$ is ineffective (or astronomically large when made quantitative), leaving a gap between the verified low degrees $n \le 8$ and the asymptotic high-degree regime.

Best known results

Terence Tao (2020/2022): Sendov's conjecture holds for all polynomials of degree $n \ge n_0$ for some absolute constant $n_0$ .
Brown and Xiang (1999): Sendov's conjecture holds for all polynomials of degree $2 \le n \le 8$ .
Bojanov, Rahman, and Szynal (1985): for every degree $n \ge 2$ and every zero $z_k \in \overline{\mathbb{D}}$ , there is a critical point within distance $(1 + |z_1 z_2 \cdots z_n|)^{1/n} \le 2^{1/n}$ of $z_k$ .

Tools and where they stop

Tool	Achieved	Where it stops
Grace's apolarity theorem and logarithmic derivative estimates (Dégot–Chalebgwa)	Proves the conjecture with explicit degree bounds when $\|z_k\|$ is bounded away from both $0$ and $1$	The explicit degree threshold $N(a)$ blows up as $\|z_k\| \to 0$ or $\|z_k\| \to 1$
Potential theory, balayage of measures, and compactness (Tao)	Controls the asymptotic distribution of zeros and critical points as $n \to \infty$ when $z_k \to 0$ or $\|z_k\| \to 1$	Ultrapower/compactness arguments do not yield a small enough threshold $n_0$ to check the remaining degrees computationally

Open questions

Can the high-degree threshold $n_0$ in Tao's theorem be lowered to $n_0 = 9$ , completing the proof of Sendov's conjecture for every degree $n \ge 2$ ?

References

Terence Tao (2022). Sendov's conjecture for sufficiently-high-degree polynomials · DOI:10.4310/ACTA.2022.v229.n2.a3 · arXiv:2012.04125
Johnny E. Brown, Guangping Xiang (1999). Proof of the Sendov conjecture for polynomials of degree at most eight · DOI:10.1006/jmaa.1999.6277
Jérôme Dégot (2014). Sendov conjecture for high degree polynomials · DOI:10.1090/S0002-9939-2014-11888-0