22,182 research outputs found

    Multiplicities of Noetherian deformations

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    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 ϵ\epsilon, which remains valid even when the system degenerates at ϵ=0\epsilon=0. 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

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    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

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    We give a separation bound for an isolated multiple root xx of a square multivariate analytic system ff satisfying that an operator deduced by adding Df(x)Df(x) and a projection of D2f(x)D^2f(x) in a direction of the kernel of Df(x)Df(x) is invertible. We prove that the deflation process applied on ff and this kind of roots terminates after only one iteration. When xx is only given approximately, we give a numerical criterion for isolating a cluster of zeros of ff near xx. 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

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    Consider a polynomial FF in mm variables and a finite point ensemble S=S1×...×SmS=S_1 \times ... \times S_m. When given the leading monomial of FF with respect to a lexicographic ordering we derive improved information on the possible number of zeros of FF of multiplicity at least rr from SS. 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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