The Puiseux Characteristic of a Goursat Germ

Shanbrom, Corey

doi:10.1007/s10883-013-9207-2

Cited by 4 publications

(7 citation statements)

References 8 publications

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“…Puiseux numbers and growth vectors. In [20], we gave a formula for the Puiseux characteristic of an analytic plane curve germ which represents a Goursat distribution germ with prescribed small growth vector.…”

Section: 2mentioning

confidence: 99%

“…In [23], Proposition 4.3.8 shows that it is equivalent to at least seven other classical invariants. In short, [20] provided the dashed arrow in the following diagram:…”

Section: 2mentioning

confidence: 99%

“…Each point in the Semple Tower is assigned an RVT code, and the code of a Goursat germ at a reference point p is that of p itself. See [16] for details, or Section 2.2 of [20] for a summary.…”

Section: Semple Meets Milnormentioning

confidence: 99%

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Bridges between subriemannian geometry and algebraic geometry: Now and then

Castro¹,

Howard²,

Shanbrom³

2015

Dynamical Systems and Differential Equations, AIMS Proceedings 2015 Proceedings of the 10th AIMS International Conference (Madr

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We consider how the problem of determining normal forms for a specific class of nonholonomic systems leads to various interesting and concrete bridges between two apparently unrelated themes. Various ideas that traditionally pertain to the field of algebraic geometry emerge here organically in an attempt to elucidate the geometric structures underlying a large class of nonholonomic distributions known as Goursat constraints. Among our new results is a regularization theorem for curves stated and proved using tools exclusively from nonholonomic geometry, and a computation of topological invariants that answer a question on the global topology of our classifying space. Last but not least we present for the first time some experimental results connecting the discrete invariants of nonholonomic plane fields such as the RVT code and the Milnor number of complex plane algebraic curves.

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Section: 2mentioning

confidence: 99%

“…In [23], Proposition 4.3.8 shows that it is equivalent to at least seven other classical invariants. In short, [20] provided the dashed arrow in the following diagram:…”

Section: 2mentioning

confidence: 99%

See 1 more Smart Citation

Bridges between subriemannian geometry and algebraic geometry: Now and then

Castro¹,

Howard²,

Shanbrom³

2015

Dynamical Systems and Differential Equations, AIMS Proceedings 2015 Proceedings of the 10th AIMS International Conference (Madr

Self Cite

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show abstract

“…Each point in the Semple Tower is assigned an RVT code, and the code of a Goursat germ at a reference point p is that of p itself. See [17] for details, or Section 2.2 of [22] for a summary.…”

Section: Semple Meets Milnormentioning

confidence: 99%

“…Puiseux numbers and growth vectors. In [22], we gave a formula for the Puiseux characteristic of an analytic plane curve germ which represents a Goursat distribution germ with prescribed small growth vector.…”

mentioning

confidence: 99%

Bridges Between Subriemannian Geometry and Algebraic Geometry

Castro,

Howard,

Shanbrom

2014

Preprint

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A Coarse Stratification of the Monster Tower

Castro¹,

Colley²,

Kennedy³

et al. 2017

Michigan Math. J.

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Abstract. The monster tower is a tower of spaces over a specified base; each space in the tower is a parameter space for curvilinear data up to a specified order. We describe and analyze a natural stratification of these spaces.The monster tower, also known in the algebro-geometric literature as the Semple tower, is a tower of smooth spaces (varieties) over a specified smooth base M. Each space M(k) in the tower, called a monster space, is a parameter space for curvilinear data up to order k on M. We will describe a coarse stratification of each monster space, with each stratum corresponding to a code word created out of a certain alphabet according to rules that we will specify. These strata parametrize curvilinear data "of the same type." The monster space can be regarded as an especially nice compactification of the parameter space for curvilinear data of nonsingular curves on M (as explained in Section 2), with the added points representing the data of singular curves; in our stratification the nonsingular data points will form a single open dense stratum.Versions of this coarse stratification have been observed by virtually everyone who has studied the monster construction. Here we develop the theory in full generality, beginning with a base space of arbitrary dimension and at all levels. A finer stratification would result from a thorough analysis of the orbits under the action of a suitable group acting on the base (whose action can be lifted to the tower) or, working locally, of the pseudogroup of local diffeomorphisms at a selected point. Results such as those in Section 5.7 of [12] show that one can expect there to be infinitely many strata, i.e., that there are moduli. We seem to be very far, however, from a full understanding of where and why moduli occur.

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The Puiseux Characteristic of a Goursat Germ

Cited by 4 publications

References 8 publications

Bridges between subriemannian geometry and algebraic geometry: Now and then

Bridges between subriemannian geometry and algebraic geometry: Now and then

Bridges Between Subriemannian Geometry and Algebraic Geometry

A Coarse Stratification of the Monster Tower

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