ThmDex – An index of mathematical definitions, results, and conjectures.
Sendov's conjecture

Let $z \mapsto f(z) : = \sum_{n = 0}^N a_n z^n$ be a D4312: Complex polynomial function such that
 (i) $$\forall \, z \in \mathbb{C} \left( f(z) = 0 \quad \implies \quad |z| \leq 1 \right)$$ (ii) $$\mathsf{Ker}(f) : = \{ z \in \mathbb{C} : f(z) = 0 \}$$
Then $$\forall \, z \in \mathbb{C} \left( f'(z) = 0 \quad \implies \quad \exists \, z_0 \in \mathsf{Ker}(f) : |z - z_0| \leq 1 \right)$$

Let $z \mapsto f(z) : = \sum_{n = 0}^N a_n z^n$ be a D4312: Complex polynomial function such that
 (i) $$\mathsf{Ker}(f) : = \{ z \in \mathbb{C} : f(z) = 0 \}$$ (ii) $$\mathsf{Ker}(f) \subseteq B[0, 1]$$
Then $$\forall \, z \in \mathbb{C} \left( f'(z) = 0 \quad \implies \quad \exists \, z_0 \in \mathsf{Ker}(f) : z \in B[z_0, 1] \right)$$

Let $z \mapsto f(z) : = \sum_{n = 0}^N a_n z^n$ be a D4312: Complex polynomial function such that
 (i) $$\mathsf{Ker}(f) : = \{ z \in \mathbb{C} : f(z) = 0 \}$$ (ii) $$\mathsf{Ker}(f') : = \{ z \in \mathbb{C} : f'(z) = 0 \}$$ (iii) $$\mathsf{Ker}(f) \subseteq B[0, 1]$$
Then $$\forall \, z \in \mathbb{C} \left( z \in \mathsf{Ker}(f') \quad \implies \quad \exists \, z_0 \in \mathsf{Ker}(f) : z \in B[z_0, 1] \right)$$