Puiseux series

In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate T. They were first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850. For example, the series In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate T. They were first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850. For example, the series is a Puiseux series in T. Puiseux's theorem, sometimes also called the Newton–Puiseux theorem, asserts that, given a polynomial equation P ( x , y ) = 0 {displaystyle P(x,y)=0} , its solutions in y, viewed as functions of x, may be expanded as Puiseux series that are convergent in some neighbourhood of the origin (0 excluded, in the case of a solution that tends to infinity at the origin). In other words, every branch of an algebraic curve may be locally (in terms of x) described by a Puiseux series. 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 Laurent series. This statement is also referred to as Puiseux's theorem, being an expression of the original Puiseux theorem in modern abstract language. Puiseux series are generalized by Hahn series. If K is a field (such as the complex numbers) then we can define the field of Puiseux series with coefficients in K informally as the set of expressions of the form where n {displaystyle n} is a positive integer and k 0 {displaystyle k_{0}} is an arbitrary integer. In other words, Puiseux series differ from Laurent series in that they allow for fractional exponents of the indeterminate, as long as these fractional exponents have bounded denominator (here n). Just as with Laurent series, Puiseux series allow for negative exponents of the indeterminate as long as these negative exponents are bounded below (here by k 0 {displaystyle k_{0}} ). Addition and multiplication are as expected: for example,