Related papers: Lojasiewicz inequality over the ring of power seri…
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…
We give here a counter-example to a conjecture of Spivakovsky. M. Spivakovsky conjectured that the function that appears in the strong Artin approximation theorem is bounded by a linear function. First we show that there is no Liouville…
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…
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…
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…
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 suggest the necessary/sufficient criteria for the existence of a (order-by-order) solution y(x) of a functional equation F(x,y)=0 over a ring. In full generality, the criteria hold in the category of filtered groups, this includes the…
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…
In this paper we derive a new strong convergence theorem of Riesz logarithmic means of the one-dimensional Vilenkin-Fourier (Walsh-Fourier) series. The corresponding inequality is pointed out and it is also proved that the inequality is in…
This note presents three resonances in commutative algebra and analytic geometry of the concept of Lojasiewicz inequality. The first is the interpretation in complex analytic geometry of the best possible exponent for a function g with…
In this paper we observe that the {\L}ojasiewicz exponent $\mathcal{L}_0(X)$ of an ADE-type singularity $X$ can be computed by means of invariants of certain ideals in the local ring ${\mathcal O}_{X,0}$. After extending the notion of…
The Polyak-{\L}ojasiewicz (P{\L}) inequality extends the favorable optimization properties of strongly convex functions to a broader class of functions. In this paper, we prove a theorem (also obtained by Criscitiello, Rebjock and Boumal in…
We prove a continuity property in the sense of currents of a continuous family of holomorphic functions which allows us to obtain a \L ojasiewicz inequality with an effective exponent independent of the parameter.
We develop a unified algebraic and valuative theory of Lojasiewicz exponents for pairs of graded families and filtrations of ideals. Within this framework, local Lojasiewicz exponents, gradient exponents, and exponents at infinity are all…
We consider formal power series in several variables with coefficients in arbitrary field such that their Newton polyhedron has a loose edge. We show that if the symbolic restriction of the power series $f$ to such an edge is a product of…
We prove a power series ring analogue of the Dedekind-Mertens lemma. Along the way, we give limiting counterexamples, we note an application to integrality, and we correct an error in the literature.
We extend a factorization theorem by Gwo\'zdziewicz and Hejmej from the ring of formal power series to any complete regular local ring $ R $. More precisely, let $ f \in R $ and assume that its Newton polyhedron has a loose edge such that…
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 consider \L ojasiewicz inequalities for a non-degenerate holomorphic function with an isolated singularity at the origin. We give an explicit estimation of the \L ojasiewicz exponent in a slightly weaker form than the assertion in…
In this paper, we prove several results on the finiteness of local cohomology of polynomial and formal power series rings. In particular, we give a partial affirmative answer for a question of L. N\'{u}\~{n}ez-Betancourt in [J. Algebra 399…