• The first one is the Jacobian conjecture: We have polynomials $$f_1, \ldots, f_n$$ in the variables $$x_1, \ldots, x_n$$ with coefficients in a field $$k$$ of non-zero characteristic. We define a function $$F\colon k^n \to k^n$$ by setting $F(x_1, \ldots, x_n) := (f_1(x_1, \ldots, x_n), \ldots, f_n(x_1, \ldots, x_n))\,.$ The conjecture now states that if the Jacobian determinant of $$F$$, which is itself a polynomial function in the variables $$x_1, \ldots, x_n$$, is a non-zero constant, then $$F$$ admits a polynomial inverse function.
• The second is Sendov’s conjecture: It states that if we have a polynomial $$f(z) = (z-r_1) \cdots (z-r_n)$$ whose roots $$r_1, \ldots, r_n$$ all lie in the closed unit disc, then each one of the roots is at a distance no more than $$1$$ from at least one critical point of $$f$$.