Not all elementary functions can be expressed with exp-minus-log
6 hours ago
- #monodromy groups
- #EML terms
- #elementary functions
- The paper 'All Elementary Functions from a Single Operator' claims that EML terms (exp x - log y) can express all elementary functions, but this depends on a specific definition of 'elementary'.
- Standard mathematical definitions of elementary functions include polynomial roots, which EML terms cannot fully express due to limitations in solvable monodromy groups.
- Using Khovanskii's topological Galois theory, it is shown that EML terms have solvable monodromy groups, while functions like roots of generic quintics have non-solvable groups (e.g., S5), proving EML terms are a strict subset of standard elementary functions.
- The article clarifies that EML terms are not universal like Boolean NAND gates, as they fail to capture key aspects of standard elementary functions, such as arbitrary polynomial roots.