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We show that three problems involving linear difference equations with rational function coefficients are essentially equivalent. The first problem is the generalization of the classical Skolem–Mahler–Lech theorem to rational function coefficients. The second problem is whether or not for a given linear difference equation there exists a Picard–Vessiot extension inside the ring of sequences. The third problem is a certain special case of the dynamical Mordell–Lang conjecture. This allows us to deduce solutions to the first two problems in a particular but fairly general special case.
Wibmer, Michael. "Skolem–Mahler–Lech type theorems and Picard–Vessiot theory." Journal of the European Mathematical Society 017.3 (2015): 523-533. <http://eudml.org/doc/277614>.
@article{Wibmer2015, abstract = {We show that three problems involving linear difference equations with rational function coefficients are essentially equivalent. The first problem is the generalization of the classical Skolem–Mahler–Lech theorem to rational function coefficients. The second problem is whether or not for a given linear difference equation there exists a Picard–Vessiot extension inside the ring of sequences. The third problem is a certain special case of the dynamical Mordell–Lang conjecture. This allows us to deduce solutions to the first two problems in a particular but fairly general special case.}, author = {Wibmer, Michael}, journal = {Journal of the European Mathematical Society}, keywords = {linear difference equations; Picard–Vessiot theory; Skolem–Mahler–Lech theorem; dynamical Mordell–Lang conjecture; linear difference equations; Picard-Vessiot theory; Skolem-Mahler-Lech theorem; dynamical Mordell-Lang conjecture}, language = {eng}, number = {3}, pages = {523-533}, publisher = {European Mathematical Society Publishing House}, title = {Skolem–Mahler–Lech type theorems and Picard–Vessiot theory}, url = {http://eudml.org/doc/277614}, volume = {017}, year = {2015}, }
TY - JOUR AU - Wibmer, Michael TI - Skolem–Mahler–Lech type theorems and Picard–Vessiot theory JO - Journal of the European Mathematical Society PY - 2015 PB - European Mathematical Society Publishing House VL - 017 IS - 3 SP - 523 EP - 533 AB - We show that three problems involving linear difference equations with rational function coefficients are essentially equivalent. The first problem is the generalization of the classical Skolem–Mahler–Lech theorem to rational function coefficients. The second problem is whether or not for a given linear difference equation there exists a Picard–Vessiot extension inside the ring of sequences. The third problem is a certain special case of the dynamical Mordell–Lang conjecture. This allows us to deduce solutions to the first two problems in a particular but fairly general special case. LA - eng KW - linear difference equations; Picard–Vessiot theory; Skolem–Mahler–Lech theorem; dynamical Mordell–Lang conjecture; linear difference equations; Picard-Vessiot theory; Skolem-Mahler-Lech theorem; dynamical Mordell-Lang conjecture UR - http://eudml.org/doc/277614 ER -