Simplification of surface parametrizations

Schicho, Josef

doi:10.1145/780506.780536

Cited by 9 publications

(7 citation statements)

References 11 publications

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“…This behavior does not hold for the case of surfaces (see [6]). Furthermore, in [7] the problem of simplifying the degree of a proper surface parametrization is analyzed. In addition a properness criteria for the surface case can be found in [5].…”

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confidence: 99%

Computation of the degree of rational surface parametrizations

Pérez–Díaz

Sendra

2004

Journal of Pure and Applied Algebra

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A rational a ne parametrization of an algebraic surface establishes a rational correspondence of the a ne plane with the surface. We consider the problem of computing the degree of such a rational map. In general, determining the degree of a rational map can be achieved by means of elimination theoretic methods. For curves, it is shown that the degree can be computed by gcd computations. In this paper, we show that the degree of a rational map induced by a surface parametrization can be computed by means of gcd and univariate resultant computations. The basic idea is to express the elements of a generic ÿbre as the ÿnitely many intersection points of certain curves directly constructed from the parametrization, and deÿned over the algebraic closure of a ÿeld of rational functions.Let P( t ) be a rational a ne parametrization of a unirational surface V over an algebraically closed ÿeld K of characteristic zero. Associated with the parametrization P( t ), we have the rational map P : K 2 → V ; t → P( t ), where P (K 2 ) ⊂ V is dense.

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confidence: 99%

Computation of the degree of rational surface parametrizations

Pérez–Díaz

Sendra

2004

Journal of Pure and Applied Algebra

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“…By resolving the base points of the parametrization, we get a resolutionS of the singularities of S. Explicit analysis of the base points (see [16]) shows that there is one base point of multiplicity n 2 − 2n, with n−3 2 infinitely near base points of multiplicity 2n and 2n + 3 infinitely near base points of multiplicity n, and there is also one simple base point with n 2 − 2n − 1 infinitely near simple base points.…”

Section: Examplementioning

confidence: 99%

The parametric degree of a rational surface

Schicho

2006

Math. Z.

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The parametric degree of a rational surface is the degree of the polynomials in the smallest possible proper parametrization. An example shows that the parametric degree is not a geometric but an arithmetic concept, in the sense that it depends on the choice of the ground field. In this paper, we introduce two geometric invariants of a rational surface, namely level and keel. These two numbers govern the parametric degree in the sense that there exist linear upper and lower bounds.

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“…Some examples are, for instance, the study of the degree of a rational surface by means of the degree of the polynomials in its rational parametrizations (see e.g. [9,13,18,19,25,27]), or the surjective cover of a surface by means of the images of finitely many rational parametrizations (see e.g. [4,21,22]).…”

Section: Introductionmentioning

confidence: 99%

On the base point locus of surface parametrizations: formulas and consequences

Cox¹,

Pérez–Díaz²,

Sendra³

2020

Preprint

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This paper shows that the multiplicity of the base points locus of a projective rational surface parametrization can be expressed as the degree of the content of a univariate resultant. As a consequence, we get a new proof of the degree formula relating the degree of the surface, the degree of the parametrization, the base points multiplicity, and the degree of the rational map induced by the parametrization. In addition, we extend both formulas to the case of dominant rational maps of the projective plane and describe how the base point loci of a parametrization and its reparametrizations are related. As an application of these results, we explore how the degree of a surface reparametrization is affected by the presence of base points.

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Simplification of surface parametrizations

Cited by 9 publications

References 11 publications

Computation of the degree of rational surface parametrizations

Computation of the degree of rational surface parametrizations

The parametric degree of a rational surface

On the base point locus of surface parametrizations: formulas and consequences

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