Puiseux series named after Victor Puiseux
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    {\displaystyle {\begin{aligned}x^{-2}&+2x^{-1/2}+x^{1/3}+2x^{11/6}+x^{8/3}+x^{5}+\cdots \\&=x^{-12/6}+2x^{-3/6}+x^{2/6}+2x^{11/6}+x^{16/6}+x^{30/6}+\cdots \end{aligned}}}
  is a Puiseux series in the indeterminate x.  Puiseux series were first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850.The definition of a Puiseux series includes that the denominators of the exponents must be bounded. So, by reducing exponents to a common denominator n, a Puiseux series becomes a Laurent series in an nth root of the indeterminate. For example, the example above is a Laurent series in 
  
    
      
        
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    {\displaystyle x^{1/6}.}
   Because a complex number has n nth roots, a convergent Puiseux series typically defines n functions in a neighborhood of 0.
Puiseux's theorem, sometimes also called the Newton–Puiseux theorem, asserts that, given a polynomial equation 
  
    
      
        P
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    {\displaystyle P(x,y)=0}
   with complex coefficients, its solutions in y, viewed as functions of x, may be expanded as Puiseux series in x that are convergent in some neighbourhood of 0. In other words, every branch of an algebraic curve may be locally described by a Puiseux series in x (or in x − x0 when considering branches above a neighborhood of x0 ≠ 0).
Using modern terminology, Puiseux's theorem asserts that the set of Puiseux series over an algebraically closed field of characteristic 0 is itself an algebraically closed field, called the field of Puiseux series. It is the algebraic closure of the field of formal Laurent series, which itself is the field of fractions of the  ring of formal power series.Newton–Puiseux theorem
As early as 1671, Isaac Newton implicitly used Puiseux series and proved the following theorem for approximating with series the roots of algebraic equations whose coefficients are functions that are themselves approximated with series or polynomials. For this purpose, he introduced the Newton polygon, which remains a fundamental tool in this context. Newton worked with truncated series, and it is only in 1850 that Victor Puiseux introduced the concept of (non-truncated) Puiseux series and proved the theorem that is now known as Puiseux's theorem or Newton–Puiseux theorem. The theorem asserts that, given an algebraic equation whose coefficients are polynomials or, more generally, Puiseux series over a field of characteristic zero, every solution of the equation can be expressed as a Puiseux series. Moreover, the proof provides an algorithm for computing these Puiseux series, and, when working over the complex numbers, the resulting series are convergent.
In modern terminology, the theorem can be restated as: the field of Puiseux series over a field of characteristic zero, and the field of convergent Puiseux series over the complex numbers, are both algebraically closed.
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