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13. Rational Free Bertini


    1. Add remark

      Problem 13.1.

      [Jurij Volčič] Suppose $r$ is a noncommutative rational function and for any $\lambda\in \mathbb{C}$ and positive integer $n$, let $\mathcal{L}_{n,\lambda}(r)$ be the set of all tuples of $n\times n$ matrices such that $\det(r(X) - \lambda I_n) = 0$.

      Is it true that $\mathcal{L}_{n,\lambda}(r)$ is irreducible for all but finitely many $n\in \mathbb{Z}^+$ and $\lambda\in \mathbb{C}$ if and only if $r$ cannot be written as a composition of a (non-M{ö}bius) univariate rational function with a noncommutative rational function?
        1. Remark. The result is true for noncommutative polynomials (the free Bertini Theorem).

              Cite this as: AimPL: Noncommutative inequalities, available at http://aimpl.org/noncommineqV.