In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series
is a Puiseux series in the indeterminate x. Puiseux series were first introduced by Isaac Newton in 1676[1] and rediscovered by Victor Puiseux in 1850.[2]
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 Because a complex number has nnth 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 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.
Eure-et-Loir department Science Puiseux crater, a crater on the Moon Puiseuxseries, a mathematical series Victor Puiseux, a 19th-century French mathematician...
series (although it is a Laurent series). Similarly, fractional powers such as x 1 2 {\textstyle x^{\frac {1}{2}}} are not permitted (but see Puiseux...
{\displaystyle F[[x]]} of formal power series. Puiseuxseries Mittag-Leffler's theorem Formal Laurent series – Laurent series considered formally, with coefficients...
Victor Alexandre Puiseux (French: [pɥizø]; 16 April 1820 – 9 September 1883) was a French mathematician and astronomer. Puiseuxseries are named after...
Padé approximant Taylor series Laurent seriesPuiseuxseries Gupta 1987; Katz 1995; Roy 2021, Ch. 1. Power Series in Fifteenth-Century Kerala, pp. 1–22...
Laurent polynomials and Laurent series, the exponents of a monomial may be negative, and in the context of Puiseuxseries, the exponents may be rational...
field of the Puiseuxseries in x. Thus f may be factored in factors of the form y − P ( x ) , {\displaystyle y-P(x),} where P is a Puiseuxseries. These factors...
Hahn series (sometimes also known as Hahn–Mal'cev–Neumann series) are a type of formal infinite series. They are a generalization of Puiseuxseries (themselves...
power seriesPuiseuxseries are an extension of formal Laurent series, allowing fractional exponents Rational series Ring of restricted power series For...
field of Laurent series C ( ( t ) ) {\displaystyle \mathbb {C} (\!(t)\!)} (integer powers), or the field of (complex) Puiseuxseries C { { t } } {\displaystyle...
the field of definable numbers the field of real numbers the field of Puiseuxseries with real coefficients the Levi-Civita field the hyperreal number fields...
{\displaystyle \mathbb {C} } is isomorphic to the field of complex Puiseuxseries. However, specifying an isomorphism requires the axiom of choice. Another...
the field of formal Laurent series K ( ( t ) ) {\displaystyle K((t))} (integer powers), or the field of Puiseuxseries K { { t } } {\displaystyle K\{\{t\}\}}...
harv error: no target: CITEREFOsbourne2000 (help). See also Puiseuxseries. Power series generalize the choice of exponent in a different direction by...
form C [ [ x 1 / n ] ] {\displaystyle \mathbf {C} [[x^{1/n}]]} (cf. Puiseuxseries)[citation needed] Let B be a ring, and let A be a subring of B. Given...
semigroup, the same construction yields a semigroup ring R[G]. Free algebra Puiseuxseries Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Vol. 211...
C with respect to their usual absolute values, as well as fields of Puiseuxseries with respect to their natural valuations. Let U be an open subset of...
essentially first proved by Newton (1676), who showed the existence of Puiseuxseries for a curve from which resolution follows easily. Riemann constructed...
UnivariatePuiseuxSeries(Expression(Integer),x,0) So any coefficient may be retrieved, for instance n = 40 {\displaystyle n=40} : coefficient(exp_series,40)...
was also being investigated in France by Serret, Frenet, Bertrand and Puiseux. Here Bonnet made major contributions to the concept of curvature. In particular...
Newton–Leibniz axiom Newton–Okounkov body Newton–Pepys problem Newton–Puiseux theorem Newton fractal Newton's identities also known as Girard-Newton...