Related papers: Sur la linearite de la fonction de Artin
We prove a Lojasiewicz type inequality for a system of polynomial equations with coefficients in the ring of formal power series in two variables. This result is an effective version of the Strong Artin Approximation Theorem. From this…
We give here a result of diophantine approximation between $\O_N$, the ring of power series in several variables, and the completion of the valuation ring that dominates $\O_N$ for the $\m$-adic topology. We deduce from this that the Artin…
In 1968, M. Artin proved that any formal power series solution of a system of analytic equations may be approximated by convergent power series solutions. Motivated by this result and a similar result of P{\l}oski, he conjectured that this…
In this short note we give a proof of Liouville's theorem (every bounded entire complex function is constant) following Peterzil and Starchenko's approach to complex analysis via o-minimality.
We interpret the Artin-Rees lemma and the Izumi theorem in term of Artin function and we obtain a stable version of the Artin-Rees lemma. We present different applications of these interpretations. First we show that the Artin function of…
We give a new and elementary proof of the nested Artin approximation Theorem for linear equations with algebraic power series coefficients. Moreover, for any Noetherian local subring of the ring of formal power series, we clarify the…
The aim of this paper is to give an effective version of the Strong Artin Approximation Theorem for binomial equations. First we give an effective version of the Greenberg Approximation Theorem for polynomial equations, then using the…
In the theory of approximation there are some problems on approximation of compacts in functional spaces by nonlinear families : first we deal with the polynomial case, and then we consider the analytic case. We demonstrate a negative…
We study the ring of arithmetical functions with unitary convolution, giving an isomorphism to a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell-Everett between the ring of arithmetical…
We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce the main theorem from a variant which bounds…
The classical Liouville theorem states that a bounded harmonic function on all of $\RR^n$ must be constant. In the early 1970s, S.T. Yau vastly generalized this, showing that it holds for manifolds with nonnegative Ricci curvature.…
It was proved by Elkik that, under some smoothness conditions, the Artin functions of systems of polynomials over a Henselian pair are bounded above by linear functions. This paper gives a stronger form of this result for the class of…
We present a new, short and independent proof of the Liouville-type theorem for entire and subharmonic functions of finite order bounded outside some set of zero planar density.
We show that, under certain geometric conditions, there are no nonconstant quasiminimizers with finite $p$th power energy in a (not necessarily complete) metric measure space equipped with a globally doubling measure supporting a global…
We give here some negative results in Sturm-Liouville inverse theory, meaning that we cannot approach any of the potentials with $m+1$ integrable derivatives on $\mathbb{R}^+$ by an $\omega$-parametric analytic family better than order of…
We introduce a number field analogue of the Mertens conjecture and demonstrate its falsity for all but finitely many number fields of any given degree. We establish the existence of a logarithmic limiting distribution for the analogous…
Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. This paper investigates two typical applications:…
Liouville theorem (LT) reveals robust incompressibility of distribution function in phase space, given arbitrary potentials. However, its quantum generalization, Wigner flow, is compressible, i.e., LT is only conditionally true (e.g., for…
The Bank-Laine conjecture concerning the oscillation of solutions of second order homogeneous linear differential equations has recently been disproved by Bergweiler and Eremenko. It is shown here, however, that the conjecture is true if…
We formulate an equivariant version of Greenberg's $p$-adic Artin conjecture for smoothed equivariant $p$-adic Artin $L$-functions in the context of an arbitrary one-dimensional admissible $p$-adic Lie extension of a totally real number…