In mathematics, a Puiseux expansion is a formal power series expansion of an algebraic function. Puiseux's theorem is a classical existence theorem for such an expansion, in the case of one variable. Euclid, detail from The School of Athens by Raphael. ... In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is... In mathematics, an algebraic function of indeterminates X1, X2, ..., Xn, is a function F that satisfies some non-trivial equation P(F, X1, X2, ..., Xn) = 0, with P a polynomial in n + 1 variables over a given field K. That is, F is an implicit function that solves an algebraic... In mathematics, an existence theorem is a theorem with a statement beginning there exist(s) .., or more generally for all x, y, ... there exist(s) .... That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. ...

If K is an algebraically closed field of characteristic 0, the algebraic closure of the field of fractions of the ring In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ... In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the quotient field or the field of fractions of the integral domain. ...

K[[T]]

of formal power series in the indeterminate T can be described as the union of the formal Laurent series fields in all the fractional powers See: indeterminate (variable) statically indeterminate Division by zero This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... A Laurent series is defined with respect to a particular point c and a path of integration γ. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic. ...

T^1/n

for integers n ≥ 1 (this is not true if char(K) = p > 0). This means that locally near a point P an algebraic curve can be parametrised by a power series in some fixed T^1/n. In the interesting case when P is a singular point, there may be more than one branch. The (several) formal power series that result are called the Puiseux expansion(s), relative to P. In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ...

When the field K is the complex numbers, these Puiseux series have non-zero radius of convergence, and so provide analytic functions in terms of a fractional-power variable. Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In... In mathematics, an analytic function is a function that is locally given by a convergent power series. ...

We can also define the field of transfinite Puiseux series as follows. Take K to be any field.

Define

One can show that if K is an algebraically closed field (e.g. ), then is also an algebraically closed field, and in general it is strictly bigger than , the algebraic closure of the field of fractions of K[[T]].

The name is for Victor Puiseux (1820-1883). The theory was at least implicit in the original use of the Newton polygon. Victor Alexandre Puiseux (1820–1883) was a French Roman Catholic mathematician and astronomer. ... In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields. ...

External links

• Puiseux series at MathWorld
• Puiseux's Theorem at MathWorld
• Puiseux series at PlanetMath