Dynamical Systems seminar - On minimality and positivity for second-order holonomic sequences

Dates

Wednesday, September 22, 2021 - 14:00 to 15:00

Speaker: George Kenison (TU Wien)

An infinite sequence <u_n> of real numbers is holonomic if it satisfies a linear recurrence relation with polynomial coefficients. Such a sequence is positive if each u_n >= 0, and minimal if, given any other linearly independent sequence <v_n> satisfying the same recurrence relation, the ratio u_n/v_n converges to 0 as n tends to infinity.
In recent work, the speaker and collaborators establish a Turing reduction of the problem of deciding positivity of second-order holonomic sequences to that of deciding minimality of such sequences.  More specifically, we give a procedure for determining positivity of second-order holonomic sequences that terminates in all but an exceptional number of cases, and we show that in these exceptional cases positivity can be determined using an oracle for deciding minimality.
This is joint work with O. Klurman, E. Lefaucheux, F. Luca, P. Moree, J. Ouaknine, M.A. Whiteland, and J. Worrell.