28,681 research outputs found
Puiseux power series solutions for systems of equations
We give an algorithm to compute term-by-term multivariate Puiseux series exapansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton polygon to the tropical variety of the ideal generated by the system
An ultrametric version of the Maillet-Malgrange theorem for nonlinear q-difference equations
We prove an ultrametric q-difference version of the Maillet-Malgrange
theorem, on the Gevrey nature of formal solutions of nonlinear analytic
q-difference equations. Since \deg_q and \ord_q define two valuations on
{\mathbb C}(q), we obtain, in particular, a result on the growth of the degree
in q and the order at q of formal solutions of nonlinear q-difference
equations, when q is a parameter. We illustrate the main theorem by considering
two examples: a q-deformation of ``Painleve' II'', for the nonlinear situation,
and a q-difference equation satisfied by the colored Jones polynomials of the
figure 8 knots, in the linear case.
We consider also a q-analog of the Maillet-Malgrange theorem, both in the
complex and in the ultrametric setting, under the assumption that |q|=1 and a
classical diophantine condition.Comment: 11 pages; many language inaccuracies have been correcte
Regularity of Singular Solutions to -Poisson Equations
This work showcases level set estimates for weak solutions to the -Poisson
equation on a bounded domain, which we use to establish Lebesgue space
inclusions for weak solutions. In particular we show that if
is a bounded domain and is a weak solution to
the Dirichlet problem for Poisson's equation for with , then for
every and indeed . This result is
shown to be sharp, and similar regularity is established for solutions to the
-Poisson equation including in the edge case .Comment: 7 page
Quadratic and cubic Gaudin Hamiltonians and super Knizhnik-Zamolodchikov equations for general linear Lie superalgebras
We show that under a generic condition, the quadratic Gaudin Hamiltonians
associated to are diagonalizable on any singular
weight space in any tensor product of unitarizable highest weight
-modules. Moreover, every joint eigenbasis of the
Hamiltonians can be obtained from some joint eigenbasis of the quadratic Gaudin
Hamiltonians for the general linear Lie algebra on the
corresponding singular weight space in the tensor product of some
finite-dimensional irreducible -modules for and
sufficiently large. After specializing to , we show that similar results
hold as well for the cubic Gaudin Hamiltonians associated to
.
We also relate the set of singular solutions of the (super)
Knizhnik-Zamolodchikov equations for to the set of
singular solutions of the Knizhnik-Zamolodchikov equations for
for and sufficiently large
Representation Theory Approach to the Polynomial Solutions of q - Difference Equations : U_q(sl(3)) and Beyond,
A new approach to the theory of polynomial solutions of q - difference
equations is proposed. The approach is based on the representation theory of
simple Lie algebras and their q - deformations and is presented here for
U_q(sl(n)). First a q - difference realization of U_q(sl(n)) in terms of
n(n-1)/2 commuting variables and depending on n-1 complex representation
parameters r_i, is constructed. From this realization lowest weight modules
(LWM) are obtained which are studied in detail for the case n=3 (the well known
n=2 case is also recovered). All reducible LWM are found and the polynomial
bases of their invariant irreducible subrepresentations are explicitly given.
This also gives a classification of the quasi-exactly solvable operators in the
present setting. The invariant subspaces are obtained as solutions of certain
invariant q - difference equations, i.e., these are kernels of invariant q -
difference operators, which are also explicitly given. Such operators were not
used until now in the theory of polynomial solutions. Finally the states in all
subrepresentations are depicted graphically via the so called Newton diagrams.Comment: uuencoded Z-compressed .tar file containing two ps files
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