Termination of the Gerhold–Kauers CAD-based positivity procedure

Determine whether the Gerhold–Kauers iterative algorithm for proving positivity of P-finite sequences—specifically, the procedure that searches for an integer m such that the inductive implication u_n ≥ 0 ∧ … ∧ u_{n+m} ≥ 0 ⇒ u_{n+m+1} ≥ 0 holds and verifies this implication using cylindrical algebraic decomposition—terminates on all valid inputs defined by rational polynomial coefficients p_i ∈ Q[n] and rational initial conditions.

Background

The paper studies the Positivity Problem for P-finite sequences, where one seeks to decide whether all terms of a sequence defined by a linear recurrence with polynomial coefficients are nonnegative. Early algorithmic work by Gerhold and Kauers (2005) proposed a general method: iteratively search for a finite m such that nonnegativity of a block of consecutive terms implies nonnegativity of the next term, and discharge this implication in the existential theory of the reals using cylindrical algebraic decomposition (CAD).

While this approach successfully produced automatic proofs for many inequalities, the authors note that its global termination behavior is unresolved. This uncertainty motivates subsequent research, including methods that ensure termination under certain structural conditions (e.g., cone-based approaches). The present paper extends cone-based techniques to cases with multiple dominant eigenvalues, but the general termination of the Gerhold–Kauers procedure itself remains an open issue.