1,477 research outputs found
Characterizing Triviality of the Exponent Lattice of A Polynomial through Galois and Galois-Like Groups
The problem of computing \emph{the exponent lattice} which consists of all
the multiplicative relations between the roots of a univariate polynomial has
drawn much attention in the field of computer algebra. As is known, almost all
irreducible polynomials with integer coefficients have only trivial exponent
lattices. However, the algorithms in the literature have difficulty in proving
such triviality for a generic polynomial. In this paper, the relations between
the Galois group (respectively, \emph{the Galois-like groups}) and the
triviality of the exponent lattice of a polynomial are investigated. The
\bbbq\emph{-trivial} pairs, which are at the heart of the relations between
the Galois group and the triviality of the exponent lattice of a polynomial,
are characterized. An effective algorithm is developed to recognize these
pairs. Based on this, a new algorithm is designed to prove the triviality of
the exponent lattice of a generic irreducible polynomial, which considerably
improves a state-of-the-art algorithm of the same type when the polynomial
degree becomes larger. In addition, the concept of the Galois-like groups of a
polynomial is introduced. Some properties of the Galois-like groups are proved
and, more importantly, a sufficient and necessary condition is given for a
polynomial (which is not necessarily irreducible) to have trivial exponent
lattice.Comment: 19 pages,2 figure
Cyclic division algebras: a tool for space-time coding
Multiple antennas at both the transmitter and receiver ends of a wireless digital transmission channel may increase both data rate and reliability. Reliable high rate transmission over such channels can only be achieved through Space–Time coding. Rank and determinant code design criteria have been proposed to enhance diversity and coding gain. The special case of full-diversity criterion requires that the difference of any two distinct codewords has full rank.
Extensive work has been done on Space–Time coding, aiming at
finding fully diverse codes with high rate. Division algebras have been proposed as a new tool for constructing Space–Time codes, since they are non-commutative algebras that naturally yield linear fully diverse codes. Their algebraic properties can thus be further exploited to
improve the design of good codes.
The aim of this work is to provide a tutorial introduction to the algebraic tools involved in the design of codes based on cyclic division algebras. The different design criteria involved will be illustrated, including the constellation shaping, the information lossless property, the non-vanishing determinant property, and the diversity multiplexing trade-off. The final target is to give the complete mathematical background underlying the construction of the Golden code and the other Perfect Space–Time block codes
Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants
We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points.
Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring).
All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface.
For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system.
This yields an efficient, output-sensitive algorithm for
computing the discriminant polynomial
Sparse Gr\"obner Bases: the Unmixed Case
Toric (or sparse) elimination theory is a framework developped during the
last decades to exploit monomial structures in systems of Laurent polynomials.
Roughly speaking, this amounts to computing in a \emph{semigroup algebra},
\emph{i.e.} an algebra generated by a subset of Laurent monomials. In order to
solve symbolically sparse systems, we introduce \emph{sparse Gr\"obner bases},
an analog of classical Gr\"obner bases for semigroup algebras, and we propose
sparse variants of the and FGLM algorithms to compute them. Our prototype
"proof-of-concept" implementation shows large speed-ups (more than 100 for some
examples) compared to optimized (classical) Gr\"obner bases software. Moreover,
in the case where the generating subset of monomials corresponds to the points
with integer coordinates in a normal lattice polytope and under regularity assumptions, we prove complexity bounds which depend
on the combinatorial properties of . These bounds yield new
estimates on the complexity of solving -dim systems where all polynomials
share the same Newton polytope (\emph{unmixed case}). For instance, we
generalize the bound on the maximal degree in a Gr\"obner
basis of a -dim. bilinear system with blocks of variables of sizes
to the multilinear case: . We also propose
a variant of Fr\"oberg's conjecture which allows us to estimate the complexity
of solving overdetermined sparse systems.Comment: 20 pages, Corollary 6.1 has been corrected, ISSAC 2014, Kobe : Japan
(2014
Correlated metallic state in honeycomb lattice: Orthogonal Dirac semimetal
A novel gapped metallic state coined orthogonal Dirac semimetal is proposed
in the honeycomb lattice in terms of slave-spin representation of
Hubbard model. This state corresponds to the disordered phase of slave-spin and
has the same thermaldynamical and transport properties as usual Dirac semimetal
but its singe-particle excitation is gapped and has nontrivial topological
order due to the gauge structure. The quantum phase transition from
this orthogonal Dirac semimetal to usual Dirac semimetal is described by a
mean-field decoupling with complementary fluctuation analysis and its
criticality falls into the universality class of 2+1D Ising model while a large
anomalous dimension for the physical electron is found at quantum critical
point (QCP), which could be considered as a fingerprint of our fractionalized
theory when compared to other non-fractionalized approaches. As byproducts, a
path integral formalism for the slave-spin representation of Hubbard
model is constructed and possible relations to other approaches and the
sublattice pairing states, which has been argued to be a promising candidate
for gapped spin liquid state found in the numerical simulation, are briefly
discussed. Additionally, when spin-orbit coupling is considered, the
instability of orthogonal Dirac semimetal to the fractionalized quantum spin
Hall insulator (fractionalized topological insulator) is also expected. We hope
the present work may be helpful for future studies in slave-spin theory
and related non-Fermi liquid phases in honeycomb lattice.Comment: 12 pages,no figures, more discussions added. arXiv admin note: text
overlap with arXiv:1203.063
Zero-Temperature Phase Transitions of Antiferromagnetic Ising Model of General Spin on a Triangular Lattice
We map the ground-state ensemble of antiferromagnetic Ising model of spin-S
on a triangular lattice to an interface model whose entropic fluctuations are
proposed to be described by an effective Gaussian free energy, which enables us
to calculate the critical exponents of various operators in terms of the
stiffness constant of the interface. Monte Carlo simulations for the
ground-state ensemble utilizing this interfacial representation are performed
to study both the dynamical and the static properties of the model. This method
yields more accurate numerical results for the critical exponents. By varying
the spin magnitude in the model, we find that the model exhibits three phases
with a Kosterlitz-Thouless phase transition at 3/2<S_{KT}<2 and a locking phase
transition at 5/2 < S_L \leq 3. The phase diagram at finite temperatures is
also discussed.Comment: 15 pages, LaTeX; 10 figures in PostScript files; The revised version
appears in PRB (see Journal-ref). New electronic address of first author,
[email protected]
Implicitization of curves and (hyper)surfaces using predicted support
We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation.
For predicting the implicit support, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory.
Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial.
We implement our methods on Maple, and some on Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces.
We apply our approach to approximate implicitization,
and quantify the accuracy of the approximate output,
which turns out to be satisfactory on all tested examples; we also relate our measures to Hausdorff distance.
In building a square or rectangular matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complexes offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice.
We compare our prototype to existing software and find that it is rather competitive
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