987 research outputs found
Homotopy techniques for multiplication modulo triangular sets
International audienceWe study the cost of multiplication modulo triangular families of polynomials. Following previous work by Li et al. (2007), we propose an algorithm that relies on homotopy and fast evaluation-interpolation techniques. We obtain a quasi-linear time complexity for substantial families of examples, for which no such result was known before. Applications are given notably to additions of algebraic numbers in small characteristic
Embedding calculus knot invariants are of finite type
We show that the map on components from the space of classical long knots to
the n-th stage of its Goodwillie-Weiss embedding calculus tower is a map of
monoids whose target is an abelian group and which is invariant under clasper
surgery. We deduce that this map on components is a finite type-(n-1) knot
invariant. We also compute the second page in total degree zero for the
spectral sequence converging to the components of this tower as Z-modules of
primitive chord diagrams, providing evidence for the conjecture that the tower
is a universal finite-type invariant over the integers. Key to these results is
the development of a group structure on the tower compatible with connect-sum
of knots, which in contrast with the corresponding results for the (weaker)
homology tower requires novel techniques involving operad actions, evaluation
maps, and cosimplicial and subcubical diagrams.Comment: Revised maps to the infinitesimal mapping space model in Sections 3
and 4 and analysis of cubical diagrams in Section 5. Minor expository and
organizational changes throughout. Now 28 pages, 4 figure
The Topology of Tile Invariants
In this note we use techniques in the topology of 2-complexes to recast some
tools that have arisen in the study of planar tiling questions. With spherical
pictures we show that the tile counting group associated to a set of tiles
and a set of regions tileable by is isomorphic to a quotient of the second
homology group of a 2-complex built from . In this topological setting we
derive some well-known tile invariants, one of which we apply to the solution
of a tiling question involving modified rectangles.Comment: 25 pages, 24 figure
On the complexity of computing with zero-dimensional triangular sets
We study the complexity of some fundamental operations for triangular sets in
dimension zero. Using Las-Vegas algorithms, we prove that one can perform such
operations as change of order, equiprojectable decomposition, or quasi-inverse
computation with a cost that is essentially that of modular composition. Over
an abstract field, this leads to a subquadratic cost (with respect to the
degree of the underlying algebraic set). Over a finite field, in a boolean RAM
model, we obtain a quasi-linear running time using Kedlaya and Umans' algorithm
for modular composition. Conversely, we also show how to reduce the problem of
modular composition to change of order for triangular sets, so that all these
problems are essentially equivalent. Our algorithms are implemented in Maple;
we present some experimental results
The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-theory
Let X be a noetherian scheme of finite Krull dimension, having 2 invertible
in its ring of regular functions, an ample family of line bundles, and a global
bound on the virtual mod-2 cohomological dimensions of its residue fields. We
prove that the comparison map from the hermitian K-theory of X to the homotopy
fixed points of K-theory under the natural Z/2-action is a 2-adic equivalence
in general, and an integral equivalence when X has no formally real residue
field. We also show that the comparison map between the higher
Grothendieck-Witt (hermitian K-) theory of X and its \'etale version is an
isomorphism on homotopy groups in the same range as for the Quillen-Lichtenbaum
conjecture in K-theory. Applications compute higher Grothendieck-Witt groups of
complex algebraic varieties and rings of 2-integers in number fields, and hence
values of Dedekind zeta-functions.Comment: 17 pages, to appear in Adv. Mat
Finite subset spaces of graphs and punctured surfaces
The kth finite subset space of a topological space X is the space exp_k X of
non-empty finite subsets of X of size at most k, topologised as a quotient of
X^k. The construction is a homotopy functor and may be regarded as a union of
configuration spaces of distinct unordered points in X. We calculate the
homology of the finite subset spaces of a connected graph Gamma, and study the
maps (exp_k phi)_* induced by a map phi:Gamma -> Gamma' between two such
graphs. By homotopy functoriality the results apply to punctured surfaces also.
The braid group B_n may be regarded as the mapping class group of an
n-punctured disc D_n, and as such it acts on H_*(exp_k D_n). We prove a
structure theorem for this action, showing that the image of the pure braid
group is nilpotent of class at most floor((n-1)/2).Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-29.abs.htm
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