Complexity Bounds for the Generalization of Fulton's Intersection Multiplicity Algorithm

We analyze the Generalization of Fulton's Intersection Multiplicity Algorithm, a symbolic partial algorithm for computing intersection multiplicities via recursive algebraic rewriting. While preliminary benchmarks suggest it may perform well on certain inputs, we show the algorithm exhibits non-elementary worst-case complexity on dense systems and exponential complexity in favorable cases. These results indicate the algorithm's practical utility is confined to systems that are either sparse or of modest size. To address this, we propose an optimization that reduces the depth of recursion across branches of computation, limiting the growth responsible for non-elementary complexity.