Flatness Characterization: Two Approaches

Antritter, Felix; Lévine, Jean

doi:10.1007/978-3-642-16135-3_11

Cited by 10 publications

(2 citation statements)

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“…A very important area of applications where, according to our preliminary studies, utilization of degree-optimal moving frames is beneficial, is the control theory. In particular, the use of degreeoptimal frames can lower differential degrees of "flat outputs" (see, for instance, Polderman and Willems [36], Martin, Murray and Rouchon [33], Fabiańska and Quadrat [17], Antritter and Levine [2], Imae, Akasawa, and Kobayashi [28]). Another interesting application of algebraic frames can be found in the paper [16] by Elkadi, Galligo and Ba, devoted to the following problem: given a vector of polynomials with gcd 1, find small degree perturbations so that the perturbed polynomials have a large-degree gcd.…”

Section: Introductionmentioning

confidence: 99%

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Degree-optimal moving frames for rational curves

Hong,

Hough,

Kogan

et al. 2017

Preprint

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A moving frame at a rational curve is a basis of vectors moving along the curve. When the rational curve is given parametrically by a row vector a of univariate polynomials, a moving frame with important algebraic properties can be defined by the columns of an invertible polynomial matrix P , such that aP = [gcd(a), 0 . . . , 0]. A degree-optimal moving frame has column-wise minimal degree, where the degree of a column is defined to be the maximum of the degrees of its components. Algebraic moving frames are closely related to the univariate versions of the celebrated Quillen-Suslin problem, effective Nullstellensatz problem, and syzygy module problem. However, this paper appears to be the first devoted to finding an efficient algorithm for constructing a degree-optimal moving frame, a property desirable in various applications. We compare our algorithm with other possible approaches, based on already available algorithms, and show that it is more efficient. We also establish several new theoretical results concerning the degrees of an optimal moving frame and its components. In addition, we show that any deterministic algorithm for computing a degree-optimal algebraic moving frame can be augmented so that it assigns a degree-optimal moving frame in a GL n (K)-equivariant manner. This crucial property of classical geometric moving frames, in combination with the algebraic properties, can be exploited in various problems.

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Section: Introductionmentioning

confidence: 99%

“…Consider the row vector a in the running example (Example 5) and its associated matrix A (Example 18). Let v =[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] T . Then + 143s 3 + 194s 4 + 57s 5 + 62s 6 + 63s 7 + 42s 8 .…”

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confidence: 99%