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Linear ordinary differential equations: Breaking through the order 2 barrier. (English)
Wang, Paul S. (ed.), International symposium on Symbolic and algebraic computation 92. ISSAC 92. Berkeley, CA, USA, July 27-29, 1992. Baltimore, MD: ACM Press. 42-48 (1992).
Summary: A major subproblem for algorithms that either factor ordinary linear differential equations or compute their closed form solutions is to find their solutions $y$ which satisfy $y’/y\in\overline K(x)$ where $K$ is the constant field for the coefficients of the equation. While a decision procedure for this subproblem was known in the 19th century, it requires factoring polynomials over $\overline K$ and has not been implemented in full generality. We present here an efficient algorithm for this subproblem, which has been implemented in the AXIOM computer algebra system for equations of arbitrary order over arbitrary fields of characteristic 0. This algorithm never needs to compute with the individual complex singularities of the equation, and algebraic numbers are added only when they appear in the potential solutions. Implementation of the complete Singer algorithm for $n=2,3$ based on this building block is in progress.