Zero-Multiplicity Bounds for Holonomic Sequences

We call a sequence of integers holonomic (or P-finite) of order d if every d+1 consecutive terms satisfy a linear relation with polynomial coefficients. These provide a generalisation of C-finite sequences (commonly referred to just as linear recurrence sequences), whose corresponding recurrence relation has constant coefficients. The celebrated Skolem-Mahler-Lech theorem states that the set of zero terms of a C-finite sequence is a union of finitely many arithmetic progressions and a finite set. Subsequent work by Evertse, Schlickewei and Schmidt showed that the number of arithmetic progressions and the size of the finite set can be effectively bounded by a quantity depending only on the order d. I will speak about recent work with Jean Abou Samra, James Worrell and Joel Ouaknine on attempting to extend such zero bounds to the general holonomic case. Unfortunately, even an analogue of the Skolem-Mahler-Lech theorem is not known for general holonomic sequences. However, we show that within a subclass for which an analogue is known, namely those sequences satisfying a relation whose leading and tailing coefficients are non-vanishing mod p for some prime p, one can recover effective bounds on the number of zeros in terms of the order of the sequence, the degrees of the coefficients, and p. This constitutes the first known bounds for any non-trivial subclass of holonomic sequences past C-finite sequences.