Abstract:
D-finite power series are those whose coefficient sequences are P-recursive ("polynomially recursive"), that is they satisfy linear recurrence relations with polynomial coefficients. Algebraicity of D-finite power series is connected to (quasi-)integrality ("global boundedness") of the coefficient sequence, by a famous result due to Eisenstein [E]. However, while algebraicity of D-finite power series is now proved to be decidable (cf. previous talk), deciding integrality of P-recursive sequences is still a largely open question. This talk addresses a family of subproblems of the integrality question. I will start by revisiting the integrality criterion for the so-called "Motzkin-type sequences" due to Klazar and Luca [KL], and propose a unified approach for analyzing global boundedness and algebraicity within a broader class of P-recursive sequences. The central contribution is an algorithm that finds all algebraic solutions of certain second-order recurrence relations with linear polynomial coefficients. As algebraicity and global boundedness are shown to be equivalent in the special cases considered, the method detects all globally bounded solutions as well. This offers a systematic approach
to deciding when a given P-recursive sequence is integral or almost integral - a question that arises naturally in combinatorics and differential algebra. (Based on joint work with Alin Bostan.)

[E] G. Eisenstein, "Über eine allgemeine Eigenschaft der Reihen-Entwicklungen aller algebraischen Funktionen”. Berichte Königl.
Preuss. Akad. Wiss. Berlin, 1852, pp. 441–443.
[KL] M. Klazar and F. Luca, "On integrality and periodicity of the Motzkin numbers". Aequationes Math. 69 (2005), no. 1-2, 68–75.