91 research outputs found
Recommended from our members
Reelle Algebraische Geometrie
No full textThis workshop was organized by Michel Coste (Rennes), Claus Scheiderer (Konstanz) and Niels Schwartz (Passau). The talks focussed on recent developments in real enumerative and tropical geometry, positivity and sums of squares, real aspects of classical algebraic geometry, semialgebraic and tame geometry, and topology and singularities of real varieties
Reelle Algebraische Geometrie
No full textThis workshop was organized by Michel Coste (Rennes), Claus Scheiderer (Konstanz) and Niels Schwartz (Passau). The talks focussed on recent developments in real enumerative and tropical geometry, positivity and sums of squares, real aspects of classical algebraic geometry, semialgebraic and tame geometry, and topology and singularities of real varieties
Codes for : Two Remarks on SOS with Rational Coefficients.
No full text<p>These are the codes (Magma, Python and GAP) used to obtain the data in the paper "Two Remaks on Sums of Squares with Rational Coefficients" by Claus Scheiderer and Jose Capco. Most of the Code were used to obtain and verify examples of Galois realizable permutation groups of even degree with fixed-point-free involution satisfying property (**) in the paper (specifically the involution is identified with complex conjugation in the Galois group).</p>
<p>We write a detail on how to use and read the code in readme.txt" file.</p>
Weak Approximation for Tori over p-adic Function Fields
No full textWe study local-global questions for Galois cohomology over the function field of a curve defined over a p-adic field, the main focus being weak approximation of rational points. We construct a 9-term Poitou-Tate-type exact sequence for tori over a field as above (and also a 12-term sequence for finite modules). Like in the number field case, part of the sequence can then be used to analyze the defect of weak approximation for a torus. We also show that the defect of weak approximation is controlled by a certain subgroup of the third unramified cohomology group of the torus. © 2014 © The Author(s) 2014. Published by Oxford University Press. All rights reserved
Toric completions and bounded functions on real algebraic varieties
No full textGiven a semi-algebraic set S, we study compactifications of S that arise from embeddings into complete toric varieties. This makes it possible to describe the asymptotic growth of polynomial functions on S in terms of combinatorial data. We extend our earlier work in Plaumann and Scheiderer [‘The ring of bounded polynomials on a semi-algebraic set’, Trans. Amer. Math. Soc. 364 (2012) 4663–4682] to compute the ring of bounded functions in this setting, and discuss applications to positive polynomials and the moment problem. Complete results are obtained in special cases, like sets defined by binomial inequalities. We also show that the wild behaviour of certain examples constructed by Krug [‘Geometric interpretations of a counterexample to Hilbert's 14th problem, and rings of bounded polynomials on semialgebraic sets’, Preprint, 2011, arXiv:1105.2029] and Mondal-Netzer [‘How fast do polynomials grow on semialgebraic sets?’, J. Algebra 413 (2014) 320–344] cannot occur in a toric setting.publishe
Galois realizable permutation groups satisfying (P)
No full text<p>The included csv file (final.csv, ; as delimiter) is a delimited text file that contains data on Galois number fields such that the Galois group satisfy property (P). For more information see our paper (J. Capco and C. Scheiderer). The file lists all the transitive permutation groups of even degree 5<2d<17 with a fixed-point-free involution (fpfi) up to automorphism that satisfy (P). These involutions are always given as the product of transpositions of the form (1,2)(3,4)...(2d-1,2d) for the MAGMA generator of the group shown in the 6th column-entry of the file. This file shows that all transitive permutation group of even degree > 5 up to degree 16 with a fixed-point-free involution that satisfy (P) is Galois realizable with a complex Galois number field so that the said involution is identified with the complex conjugation. The examples are given as polynomials whose splitting field correspond to these Galois number fields.</p>
<p>Readme.txt explains how to read the csv file.</p>
<p> </p>
Tegenvoorbeelden voor het Helton-Nievermoeden
No full textHet Helton–Nievermoeden zegt dat elke convexe, semialgebraïsche verzameling semidefiniet
representeerbaar is. In december 2016 is dit vermoeden op spectaculaire wijze met
tegenvoorbeelden weerlegd door de Duitse wiskundige Claus Scheiderer. In dit artikel
beschrijft Jan Draisma de achtergronden van het vermoeden en de tegenvoorbeelden van
Scheiderer
Topologies on the subgroup lattice of a compact group
No full textAbstractLet G be a compact topological group. The lattice ΣG of its closed subgroups is algebraic in the reversed order, hence is made a compact topological semilattice by its dual Lawson topology. A second natural order-compatible compact topology on ΣG arises from the usual topology on the set of closed subsets of G. These topologies are shown to coincide precisely if the identity component is central in G, but to be essentially different otherwise, since they also fail to satisfy natural weakenings of the equality condition. In the second part of the paper the groups G are determined in which one of the lattice operations of ΣG becomes continuous with respect to either one of these topologies; several different characterizations of these cases are also provided
Semidefinite representation for convex hulls of real algebraic curves
No full textWe show that the closed convex hull of any one-dimensional semialgebraic subset of is a spectrahedral shadow, meaning that it can be written as a linear image of the solution set of some linear matrix inequality. This is proved by an application of the moment relaxation method. Given a nonsingular affine real algebraic curve and a compact semialgebraic subset of its -points, the preordering of all regular functions on that are nonnegative on is known to be finitely generated. Our main result, from which all others are derived, says that is stable, meaning that uniform degree bounds exist for weighted sum of squares representations of elements of . We also extend this last result to the case where is only virtually compact. The main technical tool for the proof of stability is the archimedean local-global principle. As a consequence of our results we show that every convex semialgebraic subset of is a spectrahedral shadow.publishe
Convex hulls of curves of genus one
No full textAbstractLet C be a real nonsingular affine curve of genus one, embedded in affine n-space, whose set of real points is compact. For any polynomial f which is nonnegative on C(R), we prove that there exist polynomials fi with f≡∑ifi2 (mod IC) and such that the degrees deg(fi) are bounded in terms of deg(f) only. Using Lasserreʼs relaxation method, we deduce an explicit representation of the convex hull of C(R) in Rn by a lifted linear matrix inequality. This is the first instance in the literature where such a representation is given for the convex hull of a nonrational variety. The same works for convex hulls of (singular) curves whose normalization is C. We then make a detailed study of the associated degree bounds. These bounds are directly related to size and dimension of the projected matrix pencils. In particular, we prove that these bounds tend to infinity when the curve C degenerates suitably into a singular curve, and we provide explicit lower bounds as well
- …
