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13. Rational Free Bertini


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      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.