Recently I stumbled upon the following two conjectures about polynomials:
- 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 \(f\).
As usual with such `elementary’ conjectures there are by now a lot of partial results, and also a lot of wrong attempts. (I actually learned about these conjectures from two such failed attempts posted to the arXiv recently.)