Symmetric Division of Linear Ordinary Differential Operators

2025-10-01

AI Summary

This paper addresses the **divisibility problem for symmetric products of linear ordinary differential operators**: given a symmetric product operator (L = operatorname{Sym}^k(M)) and one known factor (M), how to reconstruct the remaining possible factors and determine their existence. We first establish a systematic **theory of symmetric division**, grounded in differential algebra and formal solution space analysis, deriving necessary symmetry conditions and algebraic criteria for factor existence. Second, we propose a reverse decomposition approach that reformulates factor reconstruction as a symbolic computation problem under differential constraints. Third, we design and implement effective algorithms for several nontrivial cases. Our results extend the theory of differential operator factorization and provide novel tools and decision criteria for inverse problems involving symmetric structures.

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Abstract

The symmetric product of two ordinary linear differential operators $L_1,L_2$ is an operator whose solution set contains the product $f_1f_2$ of any solution $f_1$ of $L_1$ and any solution $f_2$ of~$L_2$. It is well known how to compute the symmetric product of two given operators $L_1,L_2$. In this paper we consider the corresponding division problem: given a symmetric product $L$ and one of its factors, what can we say about the other factors?

Problem

• Investigating symmetric division of linear differential operators
• Determining unknown factors from symmetric product equations
• Extending symmetric product computation to inverse division problem

Innovation

• Symmetric division of linear differential operators
• Computing unknown factors from symmetric products
• Algorithmic solution for operator factorization problem