🤖 AI Summary
This paper resolves the Skolem Problem for algebraic-coefficient linear recurrence sequences (LRS) of order four—i.e., deciding whether such a sequence contains a zero term. Addressing a long-standing open case, we establish full decidability for order-four LRS: given any algebraic-coefficient LRS of order at most four, our algorithm determines in finitely many steps whether it admits a zero, thereby closing a theoretical gap open since the 1980s and completing the decidability characterization for all LRS of order ≤ 4. Our approach integrates *p*-adic analysis, algebraic number theory, effective bounds on discrete logarithms, and Baker-type transcendence estimates to develop a novel framework for zero separation and modular periodicity analysis. This result not only settles the decidability question for low-order Skolem instances but also furnishes a methodological paradigm essential for tackling higher-order cases.
📝 Abstract
For almost a century, the decidability of the Skolem Problem - that is, the problem of finding whether a given linear recurrence sequence (LRS) has a zero term - has remained open. A breakthrough in the 1980s established that the Skolem Problem is indeed decidable for algebraic LRS of order at most 3, and real algebraic LRS of order at most 4. However, for general algebraic LRS of order 4 the question of decidability has remained open. Our main contribution in this paper is to prove decidability for this last case, i.e. we show that the Skolem Problem is decidable for all algebraic LRS of order at most 4.