22,182 research outputs found
Multiplicities of Noetherian deformations
The \emph{Noetherian class} is a wide class of functions defined in terms of
polynomial partial differential equations. It includes functions appearing
naturally in various branches of mathematics (exponential, elliptic, modular,
etc.). A conjecture by Khovanskii states that the \emph{local} geometry of sets
defined using Noetherian equations admits effective estimates analogous to the
effective \emph{global} bounds of algebraic geometry.
We make a major step in the development of the theory of Noetherian functions
by providing an effective upper bound for the local number of isolated
solutions of a Noetherian system of equations depending on a parameter
, which remains valid even when the system degenerates at
. An estimate of this sort has played the key role in the
development of the theory of Pfaffian functions, and is expected to lead to
similar results in the Noetherian setting. We illustrate this by deducing from
our main result an effective form of the Lojasiewicz inequality for Noetherian
functions.Comment: v2: reworked last section, accepted to GAF
Finite-order meromorphic solutions and the discrete Painleve equations
Let w(z) be a finite-order meromorphic solution of the second-order
difference equation w(z+1)+w(z-1) = R(z,w(z)) (eqn 1) where R(z,w(z)) is
rational in w(z) and meromorphic in z. Then either w(z) satisfies a difference
linear or Riccati equation or else equation (1) can be transformed to one of a
list of canonical difference equations. This list consists of all known
difference Painleve equation of the form (1), together with their autonomous
versions. This suggests that the existence of finite-order meromorphic
solutions is a good detector of integrable difference equations.Comment: 34 page
On isolation of singular zeros of multivariate analytic systems
We give a separation bound for an isolated multiple root of a square
multivariate analytic system satisfying that an operator deduced by adding
and a projection of in a direction of the kernel of
is invertible. We prove that the deflation process applied on and this kind
of roots terminates after only one iteration. When is only given
approximately, we give a numerical criterion for isolating a cluster of zeros
of near . We also propose a lower bound of the number of roots in the
cluster.Comment: 17 page
List decoding of a class of affine variety codes
Consider a polynomial in variables and a finite point ensemble . When given the leading monomial of with respect to
a lexicographic ordering we derive improved information on the possible number
of zeros of of multiplicity at least from . We then use this
information to design a list decoding algorithm for a large class of affine
variety codes.Comment: 11 pages, 5 table
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