Puiseux series
E627726
Puiseux series are formal power series in fractional powers of a variable, widely used in algebraic geometry and singularity theory to locally parametrize algebraic curves.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
formal power series
ⓘ
mathematical concept ⓘ |
| appearsIn |
computation of discriminants of polynomials
ⓘ
local study of plane curve singularities ⓘ study of analytic continuation of algebraic functions ⓘ |
| componentOf | local field of an algebraic curve at a branch ⓘ |
| constraint |
denominators of exponents are bounded
ⓘ
set of exponents is well-ordered ⓘ |
| definedOver | algebraically closed field of characteristic zero ⓘ |
| fieldOfStudy |
algebraic geometry
ⓘ
commutative algebra ⓘ singularity theory ⓘ valuation theory ⓘ |
| generalizationOf | formal power series in integral powers ⓘ |
| hasForm | sum a_i x^{q_i} with rational exponents q_i ⓘ |
| hasPart |
coefficients from a base field or ring
ⓘ
fractional powers of a variable ⓘ |
| hasVariant |
generalized Puiseux series
ⓘ
multivariate Puiseux series ⓘ |
| historicalDevelopment | introduced in the 19th century ⓘ |
| namedAfter | Victor Puiseux NERFINISHED ⓘ |
| property |
allows rational exponents with bounded denominators
ⓘ
can be truncated to obtain finite Puiseux polynomials ⓘ forms an algebraically closed field over an algebraically closed base field of characteristic zero ⓘ supports Cauchy product multiplication ⓘ supports termwise addition ⓘ totally ordered by exponent of the first nonzero term ⓘ |
| relatedTo |
Abhyankar valuations
NERFINISHED
ⓘ
Henselian local fields NERFINISHED ⓘ Laurent series ⓘ Newton polygon NERFINISHED ⓘ Weierstrass preparation theorem NERFINISHED ⓘ ramification of coverings ⓘ resolution of singularities of plane curves ⓘ tropical geometry via valuations ⓘ valuation on function fields ⓘ |
| usedFor |
Newton–Puiseux algorithm
NERFINISHED
ⓘ
computing intersection multiplicities ⓘ computing semigroup of values of a plane branch ⓘ describing branches of plane algebraic curves ⓘ describing monodromy of algebraic functions ⓘ expressing algebraic functions as series expansions ⓘ local parametrization of algebraic curves ⓘ resolution of curve singularities ⓘ |
| usedIn |
algorithmic algebraic geometry
ⓘ
computation of branches of polynomial equations in two variables ⓘ computer algebra systems for curve analysis ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.