Related papers: {\L}ojasiewicz ideals in Denjoy-Carleman classes
For an arithmetically Cohen--Macaulay subscheme of projective space, there is a well-known bound for the highest degree of a minimal generator for the defining ideal of the subscheme, in terms of the Hilbert function. We prove a natural…
In this paper, we focus on the associated primes of powers of monomial ideals and asymptotic behavior properties such as normally torsion-freeness, normality, the strong persistence property, and the persistence property. In particular, we…
We survey ultradifferentiable extension theorems, i.e., quantitative versions of Whitney's classical extension theorem, with special emphasis on the existence of continuous linear extension operators. The focus is on Denjoy-Carleman classes…
A classical result of Carleman, based on the theory of quasianalytic functions, shows that polynomials are dense in $L^2(\mu)$ for any $\mu$ such that the moments $\int x^k d\mu$ do not grow too rapidly as $k \to \infty$. In this work, we…
This article investigates the soft-interior and the soft-cover of operator ideals. These operations, and especially the first one, have been widely used before, but making their role explicit and analyzing their interplay with the…
In this work, we develop Fourier Analysis for a family of classes of ultradifferentiable functions of Romieu type on the torus, usually known as Denjoy-Carleman classes. Then we are able to apply our results in order to generalize some…
In this paper, we propose several calculus rules for the generalized concave Kurdyka-\L ojasiewicz (KL) property, which generalize Li and Pong's results for KL exponents. The optimal concave desingularizing function has various forms and…
Among all $C^\infty$-algebras we characterize those which are algebras of smooth functions on smooth separable Hausdorff manifolds.
The purpose of this paper is to carry out an in-depth analysis of the intriguing van Dantzig problem which consists on characterizing the set $\mathbb{D}$ of analytic characteristic functions $\mathcal{F}$ which remains stable by the action…
We introduce a generalized version of the concave Kurdyka-\L ojasiewicz (KL) property by employing nonsmooth desingularizing functions. We also present the exact modulus of the generalized concave KL property, which provides an answer to…
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…
We construct an example of a smooth convex function on the plane with a strict minimum at zero, which is real analytic except at zero, for which Thom's gradient conjecture fails both at zero and infinity. More precisely, the gradient orbits…
We devise calculus rules for the Kurdyka-\L{}ojasiewicz exponent using the rank theorem and Lie group actions. They apply to a wide class of composite and invariant functions, and are particularly suitable for handling nonisolated local…
Inspired by Bartoszy\'nski's work on small sets, we introduce a new ideal defined by interval partitions on natural numbers and summable sequences of positive reals. Similarly, we present another ideal that relies on Bartoszy\'nski's and…
Approximative properties of linear summation methods of Fourier series are considered in the Orlicz type spaces ${\mathcal S}_{M}$. In particular, in terms of approximations by such methods, constructive characteristics are obtained for…
A subideal (also called a J-ideal) is an ideal of a B(H)-ideal J. This paper is the sequel to Subideals of operators where a complete characterization of principal and then finitely generated J-ideals were obtained by first generalizing the…
Inspired by the notion of K\"onig graphs we introduce graded ideals of K\"onig type with respect to a monomial order $<$. It is shown that if $I$ is of K\"onig type, then the Cohen--Macaulay property of $\ini_<(I)$ does not depend on the…
We study the \L ojasiewicz exponent and the log canonical threshold of ideals of $\mathcal O_n$ when restricted to generic subspaces of $\mathbb C^n$ of different dimensions. We obtain effective formulas of the resulting numbers for ideals…
Let $t\mapsto A(t)$ for $t\in T$ be a $C^M$-mapping with values unbounded operators with compact resolvents and common domain of definition which are self-adjoint or normal. Here $C^M$ stands for $C^\om$ (real analytic), a quasianalytic or…