The Limits of Expressing Elementary Functions with EML Terms

Andrzej Odrzywołek's paper, 'All Elementary Functions from a Single Operator', claims that the function E(x, y) = exp x - log y, along with variables and the constant 1, can express all elementary functions. This has sparked debate, with some suggesting it could revolutionize computer engineering and machine learning. However, I find the claim intriguing but limited. Odrzywołek defines 'elementary function' using a narrow set of 36 symbols, which allows his theorem to hold under specific conditions, including modifications to the conventional log function.

In broader mathematical terms, 'elementary functions' encompass more than Odrzywołek's definition, including polynomial roots, which EML terms cannot express. I argue that EML terms, while expressive, fall short of representing standard elementary functions as understood in modern mathematics. This is because EML terms have a solvable monodromy group, which limits their ability to express certain algebraic functions.

Using Khovanskii’s topological Galois theory, I demonstrate that the monodromy group of any EML term is solvable, whereas the monodromy group of a generic quintic polynomial is not. This discrepancy shows that EML terms cannot fully capture the breadth of standard elementary functions. Thus, while Odrzywołek's work is clever, it does not equate to a universal computational tool like the Boolean NAND gate.

I also clarify the definitions of EML terms and standard elementary functions. EML terms are derived from a recursive process involving exp and log, while standard elementary functions include rational functions and are closed under arithmetic operations, composition, and algebraic adjunctions. The inability of EML terms to express polynomial roots highlights the limitations of Odrzywołek's approach.

Ultimately, this discussion is not a refutation of Odrzywołek’s work but rather an exploration of its boundaries within the broader context of mathematical definitions. The paper's title, much like mine, might be seen as clickbait, drawing attention to the nuanced debate about what constitutes an 'elementary function'.