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We present an algorithm that, given a non-rational irreducible real space curve, satisfying certain conditions, computes a rational parametrization of a space curve near the input one. For a given tolerance ∈ > 0, the algorithm c...
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We present an algorithm that, given a non-rational irreducible real space curve, satisfying certain conditions, computes a rational parametrization of a space curve near the input one. For a given tolerance ∈ > 0, the algorithm checks whether a planar projection of the given space curve is ∈-rational and, in the affirmative case, generates a planar parametrization that is lifted to a space parametrization. This output rational space curve is of the same degree as the input curve, both have the same structure at infinity, and the Hausdorff distance between their real parts is finite. Moreover, in the examples we check that the distance is small.
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The Burchnall-Chaundy problem is classical in differential algebra, seeking to describe all commutative subalgebras of a ring of ordinary differential operators whose coefficients are functions in a given class. It received less a...
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The Burchnall-Chaundy problem is classical in differential algebra, seeking to describe all commutative subalgebras of a ring of ordinary differential operators whose coefficients are functions in a given class. It received less attention when posed in the (first) Weyl algebra, namely for polynomial coefficients, while the classification of commutative subalgebras of the Weyl algebra is in itself an important open problem. Centralizers are maximal-commutative subalgebras, and we review the properties of a basis of the centralizer of an operator $L$ in normal form, following the approach of K.R. Goodearl, with the ultimate goal of obtaining such bases by computational routines. Our first step is to establish the Dixmier test, based on a lemma by J. Dixmier and the choice of a suitable filtration, to give necessary conditions for an operator $M$ to be in the centralizer of $L$. Whenever the centralizer equals the algebra generated by $L$ and $M$, we call $L$, $M$ a Burchnall-Chaundy (BC) pair. A construction of BC pairs is presented for operators of order $4$ in the first Weyl algebra. Moreover, for true rank $r$ pairs, by means of differential subresultants, we effectively compute the fiber of the rank $r$ spectral sheaf over their spectral curve.
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In this paper, we introduce the notion of rational Hausdorff divisor, we analyze the dimension and irreducibility of its associated linear system of curves, and we prove that all irreducible real curves belonging to the linear sys...
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In this paper, we introduce the notion of rational Hausdorff divisor, we analyze the dimension and irreducibility of its associated linear system of curves, and we prove that all irreducible real curves belonging to the linear system are rational and are at finite Hausdorff distance among them. As a consequence, we provide a projective linear subspace where all (irreducible) elements are solutions of the approximate parametrization problem for a given algebraic plane curve. Furthermore, we identify the linear system with a plane curve that is shown to be rational and we develop algorithms to parametrize it analyzing its fields of parametrization. Therefore, we present a generic answer to the approximate parametrization problem. In addition, we introduce the notion of Hausdorff curve, and we prove that every irreducible Hausdorff curve can always be parametrized with a generic rational parametrization having coefficients depending on as many parameters as the degree of the input curve.
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It is well known that an irreducible algebraic curve is rational (i.e. parametric) if and only if its genus is zero. In this paper, given a tolerance ∈ > 0 and an ∈-irreducible algebraic affine plane curve C of proper degree d, ...
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It is well known that an irreducible algebraic curve is rational (i.e. parametric) if and only if its genus is zero. In this paper, given a tolerance ∈ > 0 and an ∈-irreducible algebraic affine plane curve C of proper degree d, we introduce the notion of ∈-rationality, and we provide an algorithm to parametrize approximately affine ∈ -rational plane curves by means of linear systems of (d - 2)-degree curves. The algorithm outputs a rational parametrization of a rational curve C of degree d which has the same points at infinity as C. Moreover, although we do not provide a theoretical analysis, our empirical analysis shows that C and C are close in practice.
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In this paper, given two real space algebraic curves, not necessarily bounded, whose Hausdorff distance is finite, we provide bounds of their distance. These bounds are related to the distance between the projections of the space ...
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In this paper, given two real space algebraic curves, not necessarily bounded, whose Hausdorff distance is finite, we provide bounds of their distance. These bounds are related to the distance between the projections of the space curves onto a plane (say, z = 0), and the distance between the z-coordinates of points in the original curves. Therefore, we provide a theoretical result that reduces the estimation and bounding of the Hausdorff distance of algebraic curves from the spatial to the planar case. Using these results we provide an estimation method for bounding the Hausdorff distance between two space curves and we check in applications that the method is accurate and fast.
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In Pérez-Díaz et al. (2009) [5], the authors present an algorithm to parametrize approximately e -rational curves, and they show that the Hausdorff distance, w.r.t. the Euclidean distance, between the input and output curves is ...
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In Pérez-Díaz et al. (2009) [5], the authors present an algorithm to parametrize approximately e -rational curves, and they show that the Hausdorff distance, w.r.t. the Euclidean distance, between the input and output curves is finite. In this paper, we analyze this distance for a family of curves randomly generated and we empirically find a reasonable upper bound of the Hausdorff distance between each input and output curve of the family.
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The following bibliographic data, listed in the above paper, is incorrect: S.L. Rueda and J.F. Sendra , Implicitization of DPPEs and Differential Resultants. Le Matematiche, Vol. LXIII (2008)-Fasc. I, 69-71. Thus the following two...
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The following bibliographic data, listed in the above paper, is incorrect: S.L. Rueda and J.F. Sendra , Implicitization of DPPEs and Differential Resultants. Le Matematiche, Vol. LXIII (2008)-Fasc. I, 69-71. Thus the following two sentences, citing the incorrect bibliographic data, should be ignored: In Section 1: We defined the implicit equation of a system of n differential rational parametric equations in n - 1 differential parameters in Rueda and Sendra (2008). In Section 2: We introduced the notion of implicit equation in Rueda and Sendra (2008) and we include it here for completion.
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Given a system P of n linear ordinary differential polynomial parametric equations (linear DPPEs) in n-1 differential parameters, we proved in that if nonzero a differential resultant gives the implicit equation of P. Differential...
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Given a system P of n linear ordinary differential polynomial parametric equations (linear DPPEs) in n-1 differential parameters, we proved in that if nonzero a differential resultant gives the implicit equation of P. Differential resultants often vanish under specialization, which prevented us from giving an implicitization algorithm in . Motivated by Canny's method and its generalizations we consider now a linear perturbation of P and use it to give an algorithm to decide if the dimension of the implicit ideal of P is n-1 and in the affirmative case obtain the implicit equation of P. This poster presentation will contain this development together with examples illustrating the results. An extended version of this work can be found in.
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