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Fri, 07 Mar 2014 09:26:06 +0200Fri, 07 Mar 2014 09:26:06 +0200Algorithmic aspects of tropical intersection theory
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3823
In the first part of this thesis we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus is the study of moduli spaces, where the underlying combinatorics of the varieties involved allow a much more efficient way of computing certain tropical cycles. The algorithms discussed here have been implemented in an extension for polymake, a software for polyhedral computations.
In the second part we apply the algorithmic toolkit developed in the first part to the study of tropical double Hurwitz cycles. Hurwitz cycles are a higher-dimensional generalization of Hurwitz numbers, which count covers of \(\mathbb{P}^1\) by smooth curves of a given genus with a certain fixed ramification behaviour. Double Hurwitz numbers provide a strong connection between various mathematical disciplines, including algebraic geometry, representation theory and combinatorics. The tropical cycles have a rather complex combinatorial nature, so it is very difficult to study them purely "by hand". Being able to compute examples has been very helpful
in coming up with theoretical results. Our main result states that all marked and unmarked Hurwitz cycles are connected in codimension one and that for a generic choice of simple ramification points the marked cycle is a multiple of an irreducible cycle. In addition we provide computational examples to show that this is the strongest possible statement.Simon Hampedoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3823Thu, 03 Jul 2014 09:26:06 +0200Numerical Algorithms in Algebraic Geometry with Implementation in Computer Algebra System SINGULAR
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2812
Polynomial systems arise in many applications: robotics, kinematics, chemical kinetics,
computer vision, truss design, geometric modeling, and many others. Many polynomial
systems have solutions sets, called algebraic varieties, having several irreducible
components. A fundamental problem of the numerical algebraic geometry is to decompose
such an algebraic variety into its irreducible components. The witness point sets are
the natural numerical data structure to encode irreducible algebraic varieties.
Sommese, Verschelde and Wampler represented the irreducible algebraic decomposition of
an affine algebraic variety \(X\) as a union of finite disjoint sets \(\cup_{i=0}^{d}W_i=\cup_{i=0}^{d}\left(\cup_{j=1}^{d_i}W_{ij}\right)\) called numerical irreducible decomposition. The \(W_i\) correspond to the pure i-dimensional components, and the \(W_{ij}\) represent the i-dimensional irreducible components. The numerical irreducible decomposition is implemented in BERTINI.
We modify this concept using partially GrÃ¶bner bases, triangular sets, local dimension, and
the so-called zero sum relation. We present in the second chapter the corresponding
algorithms and their implementations in SINGULAR. We give some examples and timings,
which show that the modified algorithms are more efficient if the number of variables is not
too large. For a large number of variables BERTINI is more efficient.
Leykin presented an algorithm to compute the embedded components of an algebraic variety
based on the concept of the deflation of an algebraic variety.
Depending on the modified algorithm mentioned above, we will present in the third chapter an
algorithm and its implementation in SINGULAR to compute the embedded components.
The irreducible decomposition of algebraic varieties allows us to formulate in the fourth
chapter some numerical algebraic algorithms.
In the last chapter we present two SINGULAR libraries. The first library is used to compute
the numerical irreducible decomposition and the embedded components of an algebraic variety.
The second library contains the procedures of the algorithms in the last Chapter to test
inclusion, equality of two algebraic varieties, to compute the degree of a pure i-dimensional
component, and the local dimension.
Shawki Al-Rasheddoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2812Tue, 06 Dec 2011 11:23:01 +0000