208 research outputs found
List Decoding Algorithm based on Voting in Groebner Bases for General One-Point AG Codes
We generalize the unique decoding algorithm for one-point AG codes over the
Miura-Kamiya Cab curves proposed by Lee, Bras-Amor\'os and O'Sullivan (2012) to
general one-point AG codes, without any assumption. We also extend their unique
decoding algorithm to list decoding, modify it so that it can be used with the
Feng-Rao improved code construction, prove equality between its error
correcting capability and half the minimum distance lower bound by Andersen and
Geil (2008) that has not been done in the original proposal except for
one-point Hermitian codes, remove the unnecessary computational steps so that
it can run faster, and analyze its computational complexity in terms of
multiplications and divisions in the finite field. As a unique decoding
algorithm, the proposed one is empirically and theoretically as fast as the BMS
algorithm for one-point Hermitian codes. As a list decoding algorithm,
extensive experiments suggest that it can be much faster for many moderate
size/usual inputs than the algorithm by Beelen and Brander (2010). It should be
noted that as a list decoding algorithm the proposed method seems to have
exponential worst-case computational complexity while the previous proposals
(Beelen and Brander, 2010; Guruswami and Sudan, 1999) have polynomial ones, and
that the proposed method is expected to be slower than the previous proposals
for very large/special inputs.Comment: Accepted for publication in J. Symbolic Computation. LaTeX2e
article.cls, 42 pages, 4 tables, no figures. Ver. 6 added an illustrative
example of the algorithm executio
A Direttissimo Algorithm for Equidimensional Decomposition
We describe a recursive algorithm that decomposes an algebraic set into
locally closed equidimensional sets, i.e. sets which each have irreducible
components of the same dimension. At the core of this algorithm, we combine
ideas from the theory of triangular sets, a.k.a. regular chains, with Gr\"obner
bases to encode and work with locally closed algebraic sets. Equipped with
this, our algorithm avoids projections of the algebraic sets that are
decomposed and certain genericity assumptions frequently made when decomposing
polynomial systems, such as assumptions about Noether position. This makes it
produce fine decompositions on more structured systems where ensuring
genericity assumptions often destroys the structure of the system at hand.
Practical experiments demonstrate its efficiency compared to state-of-the-art
implementations
New Results on Triangulation, Polynomial Equation Solving and Their Application in Global Localization
This thesis addresses the problem of global localization from images. The overall goal is to find the location and the direction of a camera given an image taken with the camera relative a 3D world model. In order to solve the problem several subproblems have to be handled. The two main steps for constructing a system for global localization consist of model building and localization. For the model construction phase we give a new method for triangulation that guarantees that the globally optimal position is attained under the assumption of Gaussian noise in the image measurements. A common framework for the triangulation of points, lines and conics is presented. The second contribution of the thesis is in the field of solving systems of polynomial equations. Many problems in geometrical computer vision lead to computing the real roots of a system of polynomial equations, and several such geometry problems appear in the localization problem. The method presented in the thesis gives a significant improvement in the numerics when Gröbner basis methods are applied. Such methods are often plagued by numerical problems, but by using the fact that the complete Gröbner basis is not needed, the numerics can be improved. In the final part of the thesis we present several new minimal, geometric problems that have not been solved previously. These minimal cases make use of both two and three dimensional correspondences at the same time. The solutions to these minimal problems form the basis of a localization system which aims at improving robustness compared to the state of the art
Combinatorial resultants in the algebraic rigidity matroid
Motivated by a rigidity-theoretic perspective on the Localization Problem in 2D, we develop an algorithm for computing circuit polynomials in the algebraic rigidity matroid CMn associated to the Cayley-Menger ideal for n points in 2D. We introduce combinatorial resultants, a new operation on graphs that captures properties of the Sylvester resultant of two polynomials in the algebraic rigidity matroid. We show that every rigidity circuit has a construction tree from K4 graphs based on this operation. Our algorithm performs an algebraic elimination guided by the construction tree, and uses classical resultants, factorization and ideal membership. To demonstrate its effectiveness, we implemented our algorithm in Mathematica: it took less than 15 seconds on an example where a Gröbner Basis calculation took 5 days and 6 hrs
Exploiting Torus Actions: Immaculate Line Bundles on Toric Varieties and Parametrizations of Gröbner Cells
This dissertation contains two chapters on the use of torus actions in algebraic geometry.
In chapter 2 we study âimmaculate line bundlesâ on projective toric varieties. The cohomology
groups of those line bundles vanish in all degrees, including the 0-th degree. Immaculate line
bundles can be seen as building blocks of full exceptional sequences of line bundles of the variety.
All the immaculate line bundles of a toric variety X = TV(ÎŁ) can be identified in two steps.
First identify those subsets of the rays ÎŁ(1) whose geometric realization is not k-acyclic, they
will be called tempting. Those subsets of the rays give âmaculate sets/regionsâ in the class group
of the variety. A line bundle is immaculate, if it is not in any of those maculate sets. So the first
step in finding immaculate line bundles is to find all tempting subsets. When X is projective,
the main result for this is that primitive collections â subsets of the rays that do not span a
cone, but each proper subset spans a cone â are always tempting. And a subset of rays can only
be tempting if it is the union of primitive collections. The same has to hold for the complement,
too. We give descriptions of the immaculate line bundles for different examples. In particular,
we describe the immaculate locus for projective toric varieties of Picard rank 3. Most of the
results have been published in [ABKW20].
In chapter 3 we study the Hilbert scheme of n points in affine plane. It describes all ideals
in the polynomial ring of two variables whose quotient is an n-dimensional vector space. The
Hilbert scheme can be decomposed into so called Gröbner cells. They consist of those ideals
that have a prescribed leading term ideal with respect to a given term order. The Gröbner
cells for the lexicographic and the degree-lexicographic order are parametrized in [CV08] and
[Con11], respectively, by canonical Hilbert-Burch matrices. A Hilbert-Burch matrix of an ideal is
a matrix generating the syzygies of the ideal. Its maximal minors also generate the ideal. These
results are generalized in two directions. Firstly, we consider the ring of formal power series.
Here we give a parametrization of the cells that respects the Hilbert function stratification of
the punctual Hilbert scheme. In particular, this cellular decomposition restricts to a cellular
decomposition of the subscheme consisting of ideals with a prescribed Hilbert function. We use
the parametrization to describe subsets of the Gröbner cells associated to lex-segment ideals
with a given minimal number of generators. These subsets are quasi-affine varieties inside the
cell. Most of these results have been published in [HW21] and [HW23]. The second way of
changing the setting is to consider a general term order on the polynomial ring. We give a
surjection to the Gröbner cell with respect to this ordering and parametrizations of subsets of
the cell, as well as a conjecture how the parametrization of the whole cell should look like. We
also study intersections of Gröbner cells with respect to different term orders.Die vorliegende Dissertation besteht aus zwei Kapiteln zu zwei unterschiedlichen Anwendungen
von Toruswirkungen in der algebraischen Geometrie.
Die wichtigsten Objekte des Kapitels 2 sind unbefleckte GeradenbĂŒndel auf projektiven torischen VarietĂ€ten X = TV(ÎŁ), GeradenbĂŒndel, deren Kohomologiegruppen alle verschwinden.
Unbefleckte GeradenbĂŒndel können als Bausteine fĂŒr exzeptionelle Sequenzen aus GeradenbĂŒndeln dienen und somit die derivierte Kategorie der VarietĂ€t beschreiben. Die Bestimmung
von unbefleckten GeradenbĂŒndeln lĂ€sst sich in zwei Schritte aufteilen. Es lassen sich Teilmengen
der Strahlen Σ(1) des die torische VarietÀt beschreibenden FÀchers Σ identifizieren, deren
geometrische Realisierungen nicht k-azyklisch sind. Diese verlockenden Teilmengen der Strahlen
definieren befleckte Teilmengen der Klassengruppe Cl(X). Ein GeradenbĂŒndel ist genau dann
unbefleckt, wenn es in keiner befleckten Teilmenge von Cl(X) liegt. Die Bestimmung aller
unbefleckten GeradenbĂŒndel lĂ€sst sich also in zwei Schritte aufteilen. Das Bestimmen der
verlockenden Teilmengen der Strahlen und das Bestimmen der zugehörigen befleckten Regionen.
Primitive Kollektionen â Teilmengen der Strahlen, die selbst keinen Kegel des FĂ€chers aufspannen, aber jede ihrer Teilmenge spannt einen Kegel des FĂ€chers auf â sind verlockend und
auĂerdem ist eine Teilmenge nur dann verlockend, wenn sie eine Vereinigung von primitiven
Kollektionen ist. Dies muss auch fĂŒr das Komplement gelten. Wir geben die Beschreibung
fĂŒr die unbefleckten GeradenbĂŒndel fĂŒr verschiedene Beispielklassen von projektiven torischen
VarietĂ€ten. Insbesondere beschreiben wir die unbefleckten GeradenbĂŒndel fĂŒr projektive torische
VarietÀten von Picardrang 3. Die meisten dieser Ergebnisse sind in [ABKW20] erschienen.
In Kapitel 3 geht es um das Hilbertschema von n Punkten in der affinen Ebene. Seine Punkte
sind Ideale im Polynomenring k[x, y], deren Quotient ein n-dimensionaler k-Vektorraum ist.
Das Hilbertschema kann in sogenannte Gröbnerzellen unterteilt werden. Sie umfassen Ideale,
die bezĂŒglich einer Termordnung Ï ein festgelegtes Leitideal haben. In [CV08] und [Con11]
werden fĂŒr die lexikographische und gradlexikographische Termordnung Parametrisierung der
Gröbnerzellen durch kanonische Hilbert-Burch Matrizen angegeben. Hilbert-Burch Matrizen
beschreiben die Syzygien des Ideals und ihre maximalen Minoren erzeugen das Ideal. Die
Ergebnisse werden in zwei Richtungen verallgemeinert. ZunÀchst betrachten wir Ideale im Ring
der formalen Potenzreihen. Wir geben eine Parametrisierung der Zellen, bei der die lokale
Struktur der Ideale berĂŒcksichtigt wird. Insbesondere lĂ€sst sich diese zellulĂ€re Unterteilung des
lokalen Hilbertschemas auf eine zellulÀre Unterteilung des Unterschemas einschrÀnken, das nur
Ideale mit einer gegebenen Hilbertfunktion beinhaltet. Durch diese Parametrisierung lassen sich
fĂŒr Ideale in diesen Zellen kanonische Hilbert-Burch Matrizen definieren. Diese benutzen wir
um Teilmengen der Gröbnerzellen mit einer vorgegebenen minimalen Anzahl von Erzeugern zu
beschreiben. Diese Teilmengen sind quasi-affine VarietÀten in der Gröbnerzelle. Die meisten
der Resultate sind in [HW21] und [HW23] erschienen. Die zweite Möglichkeit das Setting zu
Ă€ndern, ist beliebige Termordnungen auf dem Polynomenring zu betrachten. Im zweiten Teil
von Kapitel 3 geben wir eine Surjektion auf diese Gröbnerzellen, sowie Parametrisierungen von
Teilmengen und geben eine Vermutung, wie eine Parametrisierung der ganzen Zelle aussieht.
AuĂerdem untersuchen wir Schnitte von Gröbnerzellen bezĂŒglich verschiedener Termordnungen
Associator dependent algebras and Koszul duality
We resolve a ten year old open question of Loday of describing Koszul operads
that act on the algebra of octonions. In fact, we obtain the answer by solving
a more general classification problem: we find all Koszul operads among those
encoding associator dependent algebras.Comment: 20 pages, submitted versio
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