Related papers: Equisingularity, Multiplicity, and Dependence

We establish the principle of specialization of integral dependence for submodules of finite colength of free modules, as part of the general algebraic-geometric theory of the Buchsbaum--Rim multiplicity. Then we apply the principle to the…

We study germs of analytic maps $f:(X,S)\rightarrow(\mathbb{C}^p,0)$, when $X$ is an ICIS of dimension $n<p$. We define an image Milnor number, generalizing Mond's definition, $\mu_I(X,f)$ and give results known for the smooth case such as…

We study the equisingularity of a family of function germs $\{f_t\colon(X_t,0)\to (\mathbb{C},0)\}$, where $(X_t,0)$ are $d$-dimensional isolated determinantal singularities. We define the $(d-1)$th polar multiplicity of the fibers $X_t\cap…

We study the behavior of multidegrees in families and the existence of numerical criteria to detect integral dependence. We show that mixed multiplicities of modules are upper semicontinuous functions when taking fibers and that projective…

FI-graphs were introduced by the second author and White to capture the idea of a family of nested graphs, each member of which is acted on by a progressively larger symmetric group. That work was built on the newly minted foundations of…

Let $(R, \mathfrak m)$ be a $d$-dimensional Noetherian local ring and $E$ a finitely generated $R$-submodule of a free module $R^p.$ In this work we introduce a multiplicity sequence $c_k(E), k=0,..., d+p-1$ for $E$ that generalize the…

Over a regular local ring of dimension two with maximal ideal m, we study the Buchsbaum-Rim multiplicity of a finitely generated module M of finite colength in a free module F. First, we investigate the colength of an m-primary ideal and…

In this article we prove that the Buchsbaum-Rim multiplicity $e(F/N)$ of a parameter module $N$ in a free module $F=A^r$ is bounded above by the colength $\ell_A(F/N)$. Moreover, we prove that once the equality $\ell_A(F/N)=e(F/N)$ holds…

In this work, we consider a finitely determined map germ $f$ from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^3,0)$. We characterize the Whitney equisingularity of an unfolding $F=(f_t,t)$ of $f$ through the constancy of a single invariant in the…

Let $\A$ be an irreducible Coxeter arrangement and $W$ be its Coxeter group. Then $W$ naturally acts on $\A$. A multiplicity $\bfm : \A\rightarrow \Z$ is said to be equivariant when $\bfm$ is constant on each $W$-orbit of $\A$. In this…

We continue the project of the study of reverse mathematics principles inspired by cardinal invariants. In this article in particular we focus on principles encapsulating the existence of large families of objects that are in some sense…

We define the notion of componentwise regularity and study some of its basic properties. We prove an analogue, when working with weight orders, of Buchberger's criterion to compute Gr\"obner bases; the proof of our criterion relies on a…

We give a simple way to study the isotypical components of the homology of simplicial complexes with actions of finite groups, and use it for Milnor fibers of ICIS. We study the homology of images of mappings $f_t$ that arise as…

Using invariants from commutative algebra to count geometric objects is a basic idea in singularities. For example, the multiplicity of an ideal is used to count points of intersection of two analytic sets at points of non-transverse…

We prove a new general multiplicity estimate applicable to sets of functions without any assumption on algebraic independence. The multiplicity estimates are commonly used in determining measures of algebraic independence of values of…

We give lower and upper bounds on the Buchsbaum-Rim multiplicity of finitely generated torsion-free modules over two-dimensional regular local rings, and conditions for them to attain the bounds. As consequences, we have formulae on the…

Let $(X,0)$ be an ICIS of dimension 2 and let $f:(X,0)\to (\C^2,0)$ be a map germ with an isolated instability. We look at the invariants that appear when $X_s$ is a smoothing of $(X,0)$ and $f_s:X_s\to B_\epsilon$ is a stabilization of…

We show three basic properties on the image Milnor number $\mu_I(f)$ of a germ $f\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with isolated instability. First, we show the conservation of the image Milnor number, from which one…

A new test of independence between random elements is presented in this article. The test is based on a functional of the Cram\'{e}r-von Mises type, which is applied to a $U$-process that is defined from the recurrence rates. Theorems of…

We investigate sufficient conditions for the invariance of the real Milnor number under $\mathcal{R}$-bi-Lipschitz equivalence for function-germs $ f, g \colon (\mathbb{R}^n, 0) \to (\mathbb{R}, 0) $. More generally, we explore its…