This website contains supplementary content for the mini-course on linear recurrences for Bachelor and Master students, which is part of the summer school Mathematics in Ljubljana (30.6. – 4.7.2025).

On Days 1, 2, and 3 (Tuesday, Wednesday, and Thursday) I also provide some exercises. They are optional and in general there are more of them than you can reasonably expect to solve in the limited time available during the summer school, but you can pick a subset that seems interesting to you. Generally, they are arranged from shorter/easier to longer/harder.

Lecture Materials

Day 0

Introduction

Day 1

Characterization of LRS (Exercises)

Day 2

p-adic numbers (Exercises)

Day 3

Proving the Skolem–Mahler–Lech Theorem (over the rational numbers) (Exercises)

Day 4

On decidability: Skolem meets Schanuel

The introductory lecture was for 45 minutes. Other lectures last approximately 60 minutes.

Additional Resources

• See the introduction slides (slide 22) for reference material.
• The Skolem Tool implements the algorithm from the paper Skolem meets Schanuel and can be used interactively in the browser.
• An effective description of the zeros of an LRS in positive characteristic is given in the following paper:
  Derksen. A Skolem-Mahler-Lech Theorem in positive characteristic and finite automata. Invent. Math. 168 (2007), no. 1, 175–224. arXiv:math/0510583.