Let $K$ be a field. For any valuation $\mu$ on $K[x]$ admitting key polynomials we determine the structure of the whole set of key polynomials in terms of a fixed key polynomial of minimal degree. We deduce a canonical bijection between the set of $\mu$-equivalence classes of key polynomials and the maximal spectrum of the subring of elements of degree zero in the graded algebra of $\mu$.
Partially supported by grant MTM2016-75980-P from the Spanish MEC
Enric Nart. "Key polynomials over valued fields." Publ. Mat. 64 (1) 195 - 232, 2020. https://doi.org/10.5565/PUBLMAT6412009