266 research outputs found
Counting irreducible binomials over finite fields
We consider various counting questions for irreducible binomials over finite
fields. We use various results from analytic number theory to investigate these
questions.Comment: 11 page
Non-Abelian spin-singlet quantum Hall states: wave functions and quasihole state counting
We investigate a class of non-Abelian spin-singlet (NASS) quantum Hall
phases, proposed previously. The trial ground and quasihole excited states are
exact eigenstates of certain k+1-body interaction Hamiltonians. The k=1 cases
are the familiar Halperin Abelian spin-singlet states. We present closed-form
expressions for the many-body wave functions of the ground states, which for
k>1 were previously defined only in terms of correlators in specific conformal
field theories. The states contain clusters of k electrons, each cluster having
either all spins up, or all spins down. The ground states are non-degenerate,
while the quasihole excitations over these states show characteristic
degeneracies, which give rise to non-Abelian braid statistics. Using conformal
field theory methods, we derive counting rules that determine the degeneracies
in a spherical geometry. The results are checked against explicit numerical
diagonalization studies for small numbers of particles on the sphere.Comment: 17 pages, 4 figures, RevTe
On General Off-Shell Representations of Worldline (1D) Supersymmetry
Every finite-dimensional unitary representation of the N-extended worldline
supersymmetry without central charges may be obtained by a sequence of
differential transformations from a direct sum of minimal Adinkras, simple
supermultiplets that are identifiable with representations of the Clifford
algebra. The data specifying this procedure is a sequence of subspaces of the
direct sum of Adinkras, which then opens an avenue for classification of the
continuum of so constructed off-shell supermultiplets.Comment: 21 pages, 5 illustrations; references update
A Swan-like note for a family of binary pentanomials
In this note, we employ the techniques of Swan (Pacific J. Math. 12(3):
1099-1106, 1962) with the purpose of studying the parity of the number of the
irreducible factors of the penatomial
, where is even and .
Our results imply that if , then the polynomial in
question is reducible
Efficiently and Effectively Recognizing Toricity of Steady State Varieties
We consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a multiplicative group or, more generally, a coset of a multiplicative group. For the coset case, we study the notion of shifted toric varieties which generalizes the notion of toric varieties. This requires a geometric view on the varieties rather than an algebraic view on the ideals. We present algorithms and computations on 129 models from the BioModels repository testing for group and coset structures over both the complex numbers and the real numbers. Our methods over the complex numbers are based on Gr\"obner basis techniques and binomiality tests. Over the real numbers we use first-order characterizations and employ real quantifier elimination. In combination with suitable prime decompositions and restrictions to subspaces it turns out that almost all models show coset structure. Beyond our practical computations, we give upper bounds on the asymptotic worst-case complexity of the corresponding problems by proposing single exponential algorithms that test complex or real varieties for toricity or shifted toricity. In the positive case, these algorithms produce generating binomials. In addition, we propose an asymptotically fast algorithm for testing membership in a binomial variety over the algebraic closure of the rational numbers
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