Modular algorithms in symbolic summation and symbolic integration

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This work merges two key areas in computer algebra: symbolic integration and summation, alongside fast algorithmics. In algorithmically driven fields, the analysis of algorithms, highlighted by Don Knuth's influential talk at the 1970 ICM, serves as a clear success criterion. Researchers who develop algorithms that outperform previous methods (asymptotically, in the worst case) gain immediate recognition for their contributions. However, such breakthroughs are rare, despite ongoing efforts. An alternative evaluation method involves testing new algorithms on examples, which has its limitations but can be the best available approach. George Collins, a pioneer in computer algebra, noted in 1969 that simple analysis often provides more insight than extensive empirical data, although both are valuable. Within computer algebra, some areas, like polynomial algebra and linear algebra, have adhered to this analytical methodology, while others, such as polynomial system solving, have not fully embraced it. The conventional “input size” parameters used in computer science appear insufficient, and while some natural “geometric” parameters have been identified (like solution dimension and regularity), they do not encompass all potential major advancements. Symbolic integration and summation face similar challenges in this regard.