2,518 research outputs found
Implicitization of curves and (hyper)surfaces using predicted support
We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation.
For predicting the implicit support, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory.
Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial.
We implement our methods on Maple, and some on Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces.
We apply our approach to approximate implicitization,
and quantify the accuracy of the approximate output,
which turns out to be satisfactory on all tested examples; we also relate our measures to Hausdorff distance.
In building a square or rectangular matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complexes offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice.
We compare our prototype to existing software and find that it is rather competitive
Learning Algebraic Varieties from Samples
We seek to determine a real algebraic variety from a fixed finite subset of
points. Existing methods are studied and new methods are developed. Our focus
lies on aspects of topology and algebraic geometry, such as dimension and
defining polynomials. All algorithms are tested on a range of datasets and made
available in a Julia package
Matrix representations for toric parametrizations
In this paper we show that a surface in P^3 parametrized over a 2-dimensional
toric variety T can be represented by a matrix of linear syzygies if the base
points are finite in number and form locally a complete intersection. This
constitutes a direct generalization of the corresponding result over P^2
established in [BJ03] and [BC05]. Exploiting the sparse structure of the
parametrization, we obtain significantly smaller matrices than in the
homogeneous case and the method becomes applicable to parametrizations for
which it previously failed. We also treat the important case T = P^1 x P^1 in
detail and give numerous examples.Comment: 20 page
A Polyhedral Method to Compute All Affine Solution Sets of Sparse Polynomial Systems
To compute solutions of sparse polynomial systems efficiently we have to
exploit the structure of their Newton polytopes. While the application of
polyhedral methods naturally excludes solutions with zero components, an
irreducible decomposition of a variety is typically understood in affine space,
including also those components with zero coordinates. We present a polyhedral
method to compute all affine solution sets of a polynomial system. The method
enumerates all factors contributing to a generalized permanent. Toric solution
sets are recovered as a special case of this enumeration. For sparse systems as
adjacent 2-by-2 minors our methods scale much better than the techniques from
numerical algebraic geometry
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