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Let $P_1(x),P_2(x),P_3(x),\dotsc$ be a sequence of polynomials, determined by some initial conditions and a finite-length linear recursion with coefficients being polynomials in $x$ and the index. For example, $$ P_n(x) = P_{n-1}(x) + 2xn P_{n-2}(x) + (1+x^2+n^3x) P_{n-3}(x), \quad P_1=P_2=P_3=1 $$ is such a recursion (of length 3).

Given such a recursion and initial conditions, is it a decidable problem to figure out if all polynomials in the sequence have only real roots?

If it makes a difference: let's assume all numbers appearing are rational.

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    $\begingroup$ As far as I know it's not even known whether it's decidable whether the terms are all nonzero (en.wikipedia.org/wiki/Skolem_problem), which seems like a bad sign. $\endgroup$ Commented Jan 22 at 23:34
  • $\begingroup$ @QiaochuYuan Ah, yes, I recall now the similarity. But if the coefficients are constant, there are methods to compute the limit set of zeros. $\endgroup$ Commented Jan 23 at 15:25

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