The additive–multiplicative index dichotomy conjecture

Let $f$ be a function on a finite field of characteristic $p$. Its additive index and multiplicative index measure, respectively, how closely its behavior is compatible with additive and multiplicative structure. The exceptional monomials are maps of the form

$$x\mapsto ax^{p^j},$$

where $a$ is a field element.

Additive–multiplicative index dichotomy conjecture. Apart from monomials of the form $x\mapsto ax^{p^j}$, which have additive and multiplicative indices equal to $1$, any function with low multiplicative index has high additive index and vice versa.

This conjecture expresses the heuristic that multiplicative objects cannot generally be additively special, and conversely. The source deliberately leaves “low” and “high” vague because the appropriate precise formulation and its proof appear difficult.