Fast Computation of Smith Forms of Sparse Matrices Over Local Rings

We present algorithms to compute the Smith Normal Form of matrices over two families of local rings. The algorithms use the \emph{black-box} model which is suitable for sparse and structured matrices. The algorithms depend on a number of tools, such as matrix rank computation over finite fields, for which the best-known time- and memory-efficient algorithms are probabilistic. For an \nxn matrix $A$ over the ring \Fzfe, where $f^e$ is a power of an irreducible polynomial f \in \Fz of degree $d$, our algorithm requires \bigO(\eta de^2n) operations in \F, where our black-box is assumed to require \bigO(\eta) operations in \F to compute a matrix-vector product by a vector over \Fzfe (and $\eta$ is assumed greater than \Pden). The algorithm only requires additional storage for \bigO(\Pden) elements of \F. In particular, if \eta=\softO(\Pden), then our algorithm requires only \softO(n^2d^2e^3) operations in \F, which is an improvement on known dense methods for small $d$ and $e$. For the ring \ZZ/p^e\ZZ, where $p$ is a prime, we give an algorithm which is time- and memory-efficient when the number of nontrivial invariant factors is small. We describe a method for dimension reduction while preserving the invariant factors. The time complexity is essentially linear in $\mu n r e \log p,$ where $\mu$ is the number of operations in \ZZ/p\ZZ to evaluate the black-box (assumed greater than $n$) and $r$ is the total number of non-zero invariant factors. To avoid the practical cost of conditioning, we give a Monte Carlo certificate, which at low cost, provides either a high probability of success or a proof of failure. The quest for a time- and memory-efficient solution without restrictions on the number of nontrivial invariant factors remains open. We offer a conjecture which may contribute toward that end.Comment: Preliminary version to appear at ISSAC 201

Between primitive and 2-transitive : synchronization and its friends

The second author was supported by the Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology) through the project CEMAT-CIÊNCIAS UID/Multi/ 04621/2013An automaton (consisting of a finite set of states with given transitions) is said to be synchronizing if there is a word in the transitions which sends all states of the automaton to a single state. Research on this topic has been driven by the Černý conjecture, one of the oldest and most famous problems in automata theory, according to which a synchronizing n-state automaton has a reset word of length at most (n − 1)2 . The transitions of an automaton generate a transformation monoid on the set of states, and so an automaton can be regarded as a transformation monoid with a prescribed set of generators. In this setting, an automaton is synchronizing if the transitions generate a constant map. A permutation group G on a set Ω is said to synchronize a map f if the monoid (G, f) generated by G and f is synchronizing in the above sense; we say G is synchronizing if it synchronizes every non-permutation. The classes of synchronizing groups and friends form an hierarchy of natural and elegant classes of groups lying strictly between the classes of primitive and 2-homogeneous groups. These classes have been floating around for some years and it is now time to provide a unified reference on them. The study of all these classes has been prompted by the Černý conjecture, but it is of independent interest since it involves a rich mix of group theory, combinatorics, graph endomorphisms, semigroup theory, finite geometry, and representation theory, and has interesting computational aspects as well. So as to make the paper self-contained, we have provided background material on these topics. Our purpose here is to present recent work on synchronizing groups and related topics. In addition to the results that show the connections between the various areas of mathematics mentioned above, we include a new result on the Černý conjecture (a strengthening of a theorem of Rystsov), some challenges to finite geometers (which classical polar spaces can be partitioned into ovoids?), some thoughts about infinite analogues, and a long list of open problems to stimulate further work.PostprintPeer reviewe

Nice Complete Sets of Pairwise Quasi-Orthogonal Masas:From the Basics to a Unique Encoding

In der Quantenphysik hat der Begriff der "Mutually unbiased bases", im folgenden kurz als MUBs bezeichnet, in den letzten 25 Jahren zunehmende Bedeutung erlangt. Die vorliegende Arbeit behandelt paarweise quasi-orthogonale maximale *-Unteralgebren (Masas) von Matrixalgebren, die als algebraisches Gegenstück von MUBs verstanden werden können. Neben den Grundlagen dieses Themenkomplexes werden die bekanntesten Konstruktionsverfahren sog. vollständiger Familien von MUBs vorgestellt. Standardpaare von Masas werden besonders fokussiert, zudem zu sog. "normalen Paaren" verallgemeinert. Diese passen insoweit zum bekannten Konzept der "schönen Masa-Familien" von Aschbacher et al., als dass ein normales Paar stets auch eine schöne Familie ist. Das Hauptergebnis dieser Arbeit besagt, dass alle vollständigen schönen Masa-Familien auf eine einzige Weise codiert werden können. Ein äquivalentes Ergebnis findet sich schon bei Calderbank et al.; anders als dort wird es jedoch in der vorliegenden Arbeit mithilfe elementarer Matrixalgebra hergeleitet.Mutually unbiased bases (MUBs) have gained considerable importance in quantum physics over the past 25 years. The present thesis centres on pairwise quasi-orthogonal maximal abelian ∗-subalgebras (masas) of matrix algebras, which are an algebraic counterpart of MUBs. Starting from the basics, we first discuss the connections between equivalent pictures of MUBs, and then illustrate the most famous constructions of so-called complete sets of MUBs in prime power dimensions. We attach special importance on standard pairs of masas, and generalise this notion to pairs we call normal. Our concept of normal masa pairs is compatible with nice masa families defined by Aschbacher et al., in the sense that each normal masa pair is a nice family of length two. As the main result of this thesis, we prove that one unique method permits to encode all nice complete families. Calderbank et al. had established an equivalent result earlier; by contrast to their technique, the one presented here is based on elementary matrix algebra.<br