Related papers: Frobenius-Poincar\'e function and Hilbert-Kunz mul…

Let $(R,\mathfrak m)$ be a local (Noetherian) ring of dimension $d$ and $M$ a finite length $R$-module with free resolution $G_\bullet$. De Stefani, Huneke, and N\'{u}\~{n}ez-Betancourt explored two questions about the properties of…

We show that the Hilbert-Kunz multiplicities of the reductions to positive characteristics of an irreducible projective curve in characteristic 0 have a well-defined limit as the characteristic tends to infinity.

We lay out the theory of a multiplicity in the setting of a triangulated category having a central ring action from a graded-commutative ring $R$, in other words, an $R$-linear triangulated category. The invariant we consider is modelled on…

We consider and provide an accurate study for the fractional Zernike functions on the punctured unit disc, generalizing the classical Zernike polynomials and their associated $\beta$-restricted Zernike functions. Mainly, we give the…

In this paper, we initiate a systematic study of the generalized Hilbert-Kunz multiplicity for families of ideals in a Noetherian local ring (R,m) of positive characteristic, and introduce a new asymptotic invariant called the Amao-type…

Here we study bounds on the Frobenius-Schur exponent of spherical fusion categories based on their global dimension generalizing bounds from the representation theory of finite-dimensional quasi-Hopf algebras. Our main result is that if the…

The $L$-function of exponential sums associated to the generic polynomial of degree $d$ in $n$ variables over a finite field of characteristic $p$ is studied. A polygon called the Frobenius polygon of the generic polynomial of degree $d$ in…

We establish the $L^p$ boundedness of Hilbert transforms and maximal functions along flat curves in the Heisenberg group. This generalizes the $\mathbb{R}^n$ result by Carbery, Christ, Vance, Wainger, and Watson. What is new about our…

A multidimensional generalization of the Bernstein class of functions and the properties of functions of the introduced class are examined. In particular, a new proof of the integral representation of Bernstein functions of many variables…

The asymptotic behaviour of the solutions of Poincar\'e's functional equation $f(\lambda z)=p(f(z))$ ($\lambda>1$) for $p$ a real polynomial of degree $\geq2$ is studied in angular regions of the complex plain. The constancy of an occurring…

We prove a fractional version of Poincar\'e inequalities in the context of $\R^n$ endowed with a fairly general measure. Namely we prove a control of an $L^2$ norm by a non local quantity, which plays the role of the gradient in the…

Let $K$ be a field, $A$ a standard graded $K$-algebra and $M$ a finitely generated graded $A$-module. Inspired by our previous works, we study the Hilbert depth of $h_M$, that is $$\operatorname{hdepth}(h_M)=\max\{d\;:\; \sum\limits_{j\leq…

We give examples of $L^{1}$-functions that are essentially unbounded on every nonempty open subset of their domains of definition. We obtain such functions as limits of weighted sums of functions with the unboundedly increasing number of…

We prove that admissible functions for Fubini-Study metrics on the complex projective space $P_{m}C$, of complex dimension $m$, invariant by a convenient automorphisms group, are lower bounded by a function going to minus infinity on the…

This paper concerns the question of whether a more direct limit can be used to obtain the limit Hilbert-Kunz multiplicity, a possible candidate for a characteristic zero Hilbert-Kunz multiplicity. The main goal is to establish an…

For any real number $p\in [1,+\infty)$, we characterise the operations $\mathbb{R}^I \to \mathbb{R}$ that preserve $p$-integrability, i.e., the operations under which, for every measure $\mu$, the set $\mathcal{L}^p(\mu)$ is closed. We…

Let $N$ be a positive integer and let $f$ be a meromorphic modular function of level $N$ with rational Fourier coefficients. For a prime $p$, define a function $f_p$ on the complex upper half-plane $\mathbb{H}$ by \begin{equation*}…

Let V be an infinite matrix with rows and columns indexed by the positive integers, and entries in a field F. Suppose that v_{i,j} only depends on i-j and is 0 for |i-j| large. Then V^n is defined for all n, and one has a "generating…

Consider a multiplicative function f(n) taking values on the unit circle. Is it possible that the partial sums of this function are bounded? We show that if we weaken the notion of multiplicativity so that f(pn)=f(p)f(n) for all primes p in…

Fleming's inequality is generalized to the decay function of mixed states. We show that for any symmetric hamiltonian $h$ and for any density operator $\rho$ on a finite dimensional Hilbert space with the orthogonal projection $\Pi$ onto…