26 research outputs found
Resolving zero-divisors using Hensel lifting
Algorithms which compute modulo triangular sets must respect the presence of
zero-divisors. We present Hensel lifting as a tool for dealing with them. We
give an application: a modular algorithm for computing GCDs of univariate
polynomials with coefficients modulo a radical triangular set over the
rationals. Our modular algorithm naturally generalizes previous work from
algebraic number theory. We have implemented our algorithm using Maple's RECDEN
package. We compare our implementation with the procedure RegularGcd in the
RegularChains package.Comment: Shorter version to appear in Proceedings of SYNASC 201
Discovering the roots: Uniform closure results for algebraic classes under factoring
Newton iteration (NI) is an almost 350 years old recursive formula that
approximates a simple root of a polynomial quite rapidly. We generalize it to a
matrix recurrence (allRootsNI) that approximates all the roots simultaneously.
In this form, the process yields a better circuit complexity in the case when
the number of roots is small but the multiplicities are exponentially
large. Our method sets up a linear system in unknowns and iteratively
builds the roots as formal power series. For an algebraic circuit
of size we prove that each factor has size at most a
polynomial in: and the degree of the squarefree part of . Consequently,
if is a -hard polynomial then any nonzero multiple
is equally hard for arbitrary positive 's, assuming
that is at most .
It is an old open question whether the class of poly()-sized formulas
(resp. algebraic branching programs) is closed under factoring. We show that
given a polynomial of degree and formula (resp. ABP) size
we can find a similar size formula (resp. ABP) factor in
randomized poly()-time. Consequently, if determinant requires
size formula, then the same can be said about any of its
nonzero multiples.
As part of our proofs, we identify a new property of multivariate polynomial
factorization. We show that under a random linear transformation ,
completely factors via power series roots. Moreover, the
factorization adapts well to circuit complexity analysis. This with allRootsNI
are the techniques that help us make progress towards the old open problems,
supplementing the large body of classical results and concepts in algebraic
circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \&
Burgisser, FOCS 2001).Comment: 33 Pages, No figure
The Fourier restriction and Kakeya problems over rings of integers modulo N
The Fourier restriction phenomenon and the size of Kakeya sets are explored
in the setting of the ring of integers modulo for general and a
striking similarity with the corresponding euclidean problems is observed. One
should contrast this with known results in the finite field setting
Derandomization via Symmetric Polytopes: Poly-Time Factorization of Certain Sparse Polynomials
More than three decades ago, after a series of results, Kaltofen and Trager (J. Symb. Comput. 1990) designed a randomized polynomial time algorithm for factorization of multivariate circuits. Derandomizing this algorithm, even for restricted circuit classes, is an important open problem. In particular, the case of s-sparse polynomials, having individual degree d = O(1), is very well-studied (Shpilka, Volkovich ICALP\u2710; Volkovich RANDOM\u2717; Bhargava, Saraf and Volkovich FOCS\u2718, JACM\u2720). We give a complete derandomization for this class assuming that the input is a symmetric polynomial over rationals. Generally, we prove an s^poly(d)-sparsity bound for the factors of symmetric polynomials over any field. This characterizes the known worst-case examples of sparsity blow-up for sparse polynomial factoring.
To factor f, we use techniques from convex geometry and exploit symmetry (only) in the Newton polytope of f. We prove a crucial result about convex polytopes, by introducing the concept of "low min-entropy", which might also be of independent interest
The K-theory of (compound) Du Val singularities
This thesis gives a complete description of the Grothendieck group and divisor class group for large families of two and three dimensional singularities. The main results presented throughout, and summarised in Theorem 8.1.1, give an explicit description of the Grothendieck group and class group of Kleinian singularities, their deformations, and compound Du Val (cDV) singularities in a variety of settings. For such rings R, the main results assert that there exists an isomorphism between G_0(R) and Z + Cl(R), and the class group is explicitly presented.
More precisely, we establish these results for 2-dimensional deformations of global type A Kleinian singularities, 3-dimensional isolated complete local cDV singularities admitting a noncommutative crepant resolution, any 3-dimensional type A complete local cDV singularity, polyhedral quotient singularities (which are non-isolated), and any isolated cDV singularity admitting a minimal model with only type cAn singularities. We also study various complex reflection groups in the setting of symplectic quotient singularities, for which this isomorphism does not hold, and conjecture based on computer evidence that the reduced Grothendieck group in the case of the symmetric group has size n!.
This work requires a range of tools including, but not limited to, Nagata’s theo- rem, knitting techniques, Knörrer periodicity, the singularity category, and the computer-algebra system MAGMA. Of particular note is the application of knitting techniques which leads to independently interesting results on the symmetry of the quivers underlying the modifying algebras of Kleinian and cDV singularities
Parameter Synthesis for Markov Models
Markov chain analysis is a key technique in reliability engineering. A
practical obstacle is that all probabilities in Markov models need to be known.
However, system quantities such as failure rates or packet loss ratios, etc.
are often not---or only partially---known. This motivates considering
parametric models with transitions labeled with functions over parameters.
Whereas traditional Markov chain analysis evaluates a reliability metric for a
single, fixed set of probabilities, analysing parametric Markov models focuses
on synthesising parameter values that establish a given reliability or
performance specification . Examples are: what component failure rates
ensure the probability of a system breakdown to be below 0.00000001?, or which
failure rates maximise reliability? This paper presents various analysis
algorithms for parametric Markov chains and Markov decision processes. We focus
on three problems: (a) do all parameter values within a given region satisfy
?, (b) which regions satisfy and which ones do not?, and (c)
an approximate version of (b) focusing on covering a large fraction of all
possible parameter values. We give a detailed account of the various
algorithms, present a software tool realising these techniques, and report on
an extensive experimental evaluation on benchmarks that span a wide range of
applications.Comment: 38 page
Very stable Higgs bundles, equivariant multiplicity and mirror symmetry
We define and study the existence of very stable Higgs bundles on Riemann
surfaces, how it implies a precise formula for the multiplicity of the very
stable components of the global nilpotent cone and its relationship to mirror
symmetry. The main ingredients are the Bialynicki-Birula theory of -actions on semiprojective varieties, characters of
indices of -equivariant coherent sheaves, Hecke transformation
for Higgs bundles, relative Fourier-Mukai transform along the Hitchin
fibration, hyperholomorphic structures on universal bundles and cominuscule
Higgs bundles.Comment: 91 pages, refereed version, to appear in Inventiones Mathematica