### Abstract

Formal Puiseux series are generalisations of formal power series and formal Laurent series that also allow for fractional exponents. They have the following general form: \[\sum_{i=N}^\infty a_{i/d} X^{i/d}\] where N is an integer and d is a positive integer.

This entry defines these series including their basic algebraic properties. Furthermore, it proves the Newton–Puiseux Theorem, namely that the Puiseux series over an algebraically closed field of characteristic 0 are also algebraically closed.

### License

BSD License

### Topics

- Mathematics/Algebra

### Session Formal_Puiseux_Series

- Puiseux_Polynomial_Library
- FPS_Hensel
- Formal_Puiseux_Series

### Depends on

- Polynomial Interpolation

### Used by

- Two theorems about the geometry of the critical points of a complex polynomial
- Perfect Fields

### Auto-related entries

- Perfect Fields
- Hilbert’s Nullstellensatz
- Linear Recurrences

## Cite

×

Formal_Puiseux_Series-AFP

    @article{Formal_Puiseux_Series-AFP,
      author  = {Manuel Eberl},
      title   = {Formal Puiseux Series},
      journal = {Archive of Formal Proofs},
      month   = {February},
      year    = {2021},
      note    = {\url{https://isa-afp.org/entries/Formal_Puiseux_Series.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

Download

## Download

× Download latest

Older releases:

- Sep 13, 2023 : Isabelle2023
- Oct 27, 2022 : Isabelle2022
- Dec 14, 2021 : Isabelle2021-1
- Feb 24, 2021 : Isabelle2021