Related papers: Measuring Singularities with Frobenius: The Basics
The problem "A general characterization of uniqueness polynomial for non-critically injective polynomials" has been remained open since the last two decades. In this paper, we explore this open problem. To this end, we initiate a new…
An important tool to quantify the likeness of two probability measures are f-divergences, which have seen widespread application in statistics and information theory. An example is the total variation, which plays an exceptional role among…
The problem of estimating the multiplicity of the zero of a polynomial when restricted to the trajectory of a non-singular polynomial vector field, at one or several points, has been considered by authors in several different fields. The…
The paper provides an elementary proof establishing a sharp universal bound on the $(d-1)$-Hausdorff measure of the zeros of any nontrivial multivariable polynomial $p:\mathbb{R}^d\to\mathbb{R}$ within a $d$-dimensional cube of size $r$.…
We initiate a study of the complexity of quantum field theories (QFTs) by proposing a measure of information contained in a QFT and its observables. We show that from minimal assertions, one is naturally led to measure complexity by two…
The problem of measuring an unbounded system attribute near a singularity has been discussed. Lenses have been introduced as formal objects to study increasingly precise measurements around the singularity and a specific family of lenses…
According to the classification of quasihomogeneus singularities, any polynomial $f$ defining such singularity has a decomposition $f = f_\kappa + f_{add}$. The polynomial $f_\kappa$ is of the certain form while $f_{add}$ is only restricted…
Gathering data through measurements is at the basis of every experimental science. Ideally, measurements should be repeatable and, when extracting only coarse-grained data, they should allow the experimenter to retrieve the finer details at…
A circle, centered at the origin and with radius chosen so that it has non-empty intersection with the integer lattice $\mathbb{Z}^{2}$, gives rise to a probability measure on the unit circle in a natural way. Such measures, and their weak…
In this paper, we elaborate on the connection between leading singularities and canonical bases of Feynman integrals beyond polylogarithms. We start by discussing a notion of leading singularities in dimensional regularization, which can be…
From an operational perspective, quantumness characterizes the exotic behavior in a physical process which cannot be explained with Newtonian physics. There are several widely used measures of quantumness, including coherence, discord, and…
We study finitely additive extensions of the asymptotic density to all the subsets of natural numbers. Such measures are called density measures. We consider a class of density measures constructed from free ultrafilters on $\mathbb{N}$ and…
The goal of this paper is a classification theorem of the singularities according to a new invariant, Mather discrepancy. On the other hand, we show some evidences convincing us that Mather discrepancy is a considerable invariant: By…
Inequality measures are quantitative measures that take values in the unit interval, with a zero value characterizing perfect equality. Although originally proposed to measure economic inequalities, they can be applied to several other…
Given an irreducible hypersurface singularity of dimension $d$ (defined by a polynomial $f\in K[[ {\bf x} ]][z]$) and the projection to the affine space defined by $K[[ {\bf x} ]]$, we construct an invariant which detects whether the…
Let $\mathcal{F}$ be a plane singular curve defined over a finite field $\mathbb{F}_q$. The linear system of plane curves of a given degree passing through the singularities of $\cF$ provides potentially good bounds for the number of points…
In this book we describe an approach through toric geometry to the following problem: "estimate the number (counted with appropriate multiplicity) of isolated solutions of n polynomial equations in n variables over an algebraically closed…
These lecture notes provide a unified overview of most known canonical desingularization methods in characteristic zero. It starts with discussing the classical method, and then proceeds with the recently discovered ones: logarithmic…
We extend the definitions of complexity measures of functions to domains such as the symmetric group. The complexity measures we consider include degree, approximate degree, decision tree complexity, sensitivity, block sensitivity, and a…
Watanabe's singular learning theory provides a framework for asymptotic analysis of Bayesian model selection for statistical models with singularities, where traditional statistical regularity assumptions fail. Learning coefficients, also…