Narrow Results

We prove that the images of irreducible germs of plane curves by a germ of analytic morphism φ have a certain contact either with branches of the discriminant of φ or with certain infinitesimal structures (shadows) that arise from the branches of the Jacobian of φ that are mapped to a point (and therefore give rise to no branch of the discriminant).

The Hyperspherical coordinates method (HCM), allows one to obtain the spectrum of the system and develop the classification of excited states. We have solved the radial Schrödinger wave equation exactly in three dimensions for an N-body problem. Using the obtained results, we have studied the N family baryon masses.

Ferromagnetic materials are governed by a variational principle which is nonlocal, nonconvex and multiscale. The main object is given by a unit-length three-dimensional vector field, the magnetization, that corresponds to the stable states of the micromagnetic energy. Our aim is to analyze a thin film regime that captures the asymptotic behavior of boundary vortices generated by the magnetization and their interaction energy. This study is based on the notion of “global Jacobian” detecting the topological defects that a priori could be located in the interior and at the boundary of the film. A major difficulty consists in estimating the nonlocal part of the micromagnetic energy in order to isolate the exact terms corresponding to the topological defects. We prove the concentration of the energy around boundary vortices via a $\mathrm{\Gamma }$-convergence expansion at the second order. The second-order term is the renormalized energy that represents the interaction between the boundary vortices and governs their optimal position. We compute the expression of the renormalized energy for which we prove the existence of minimizers having two boundary vortices of multiplicity $1$. Compactness results are also shown for the magnetization and the corresponding global Jacobian.

In quantum mechanics the hyper-spherical method is one of the most well-established and successful computational tools. The general theory of harmonic polynomials and hyper-spherical harmonics is of central importance in this paper. The interaction potential V is assumed to depend on the hyper-radius ρ only where ρ is the function of the Jacobi relative coordinate x₁, x₂,…, x_n which are functions of the particles' relative positions.

In the present work we give an exact analytical solution of the Schrödinger equation for an N-particle system by using the hyperspherical approach, in the presence of the hypercentral potential of form V(R) = a₁R²+b₁R^-4+c₁R^-6 for both the ground state and the excited states.

Let 𝕜 be a field of characteristic p > 0 and R be a subalgebra of 𝕜[X] = 𝕜[x₁, …, x_n]. Let J(R) be the ideal in 𝕜[X] defined by . It is shown that if it is a principal ideal then , where q = pⁿ(p - 1)/2.

For a field $k$ of characteristic $0$, we present an algorithm for deciding if a morphism $\varphi :k[X_1,\dots ,X_m]\to k[X_1,\dots ,X_m]$ has an inverse. The algorithm also shows how to find the inverse when it exists.

It is proved that the Jacobian of a k-endomorphism of k[x₁,…,x_n] over a field k of characteristic zero, taking every tame coordinate to a coordinate, must be a nonzero constant in k. It is also proved that the Jacobian of an R-endomorphism of A=:R[x₁,…,x_n] (where R is a polynomial ring in a finite number of variables over an infinite field k), taking every R-linear coordinate of A to an R-coordinate of A, is a nonzero constant in k.

We show that finite fields over which there is a curve of a given genus g ≥ 1 with its Jacobian having a small exponent, are very rare. This extends a recent result of Duke in the case of g = 1. We also show that when g = 1 or g = 2, our lower bounds on the exponent, valid for almost all finite fields 𝔽_q and all curves over 𝔽_q, are best possible.

Consider the moduli space of smooth curves of genus g and p-rank f defined over an algebraically closed field k of characteristic p. It is an open problem to classify which group schemes occur as the p-torsion of the Jacobians of these curves for f < g - 1. We prove that the generic point of every component of this moduli space has a-number 1 when f = g - 2 and f = g - 3. Likewise, we show that a generic hyperelliptic curve with p-rank g - 2 has a-number 1 when p ≥ 3. We also show that the locus of curves with p-rank g - 2 and a-number 2 is non-empty with codimension 3 in when p ≥ 5. We include some other results when f = g - 3. The proofs are by induction on g while fixing g - f. They use computations about certain components of the boundary of .

Let $K$ be a global field and let $S$ be a finite set of primes of $K$ containing the Archimedean primes. We generalize the duality theorem for the Néron $S$-class group of an abelian variety $A$ over $K$ established previously by removing the requirement that the Tate–Shafarevich group of $A$ be finite. We also derive an exact sequence that relates the indicated group associated to the Jacobian variety of a proper, smooth and geometrically connected curve $X$ over $K$ to a certain finite subquotient of the Brauer group of $X$.

In this paper, we describe a new polynomial factorization algorithm over finite fields with odd characteristics. The main ingredient of the algorithm is special singular curves. The algorithm relies on the extension of the Mumford representation and Cantor’s algorithm to these special singular curves.

We study the problem of determining, given an integer k, the rational solutions to $C_k:x^{3}z+x^2{y}^2+y^{3}z=k{z}^4$. For $k\ne 0$, the curve $C_k$ has genus 3 and its Jacobian is isogenous to the product of three elliptic curves $E_{1,k}$, $E_{2,k}$, $E_{3,k}$. We explicitly determine the rational points on $C_k$ under the assumption that one of these elliptic curves has rank zero. We discuss the challenges involved in extending our result to handle all $k\in \mathbb{Q}$.

In this work, positive functions defined on the plane are considered from a generic viewpoint, both in the continuous and bounded setting. By pursuing on constructions of Burago–Kleiner and McMullen, we show that, generically, such a function cannot be written as the Jacobian of a bi-Lipschitz homeomorphism.

J.-P. Serre asserted a precise form of Torelli Theorem for genus 3 curves, namely, an indecomposable principally polarized abelian threefold is a Jacobian if and only if some specific invariant is a square. We study here a three dimensional family of such threefolds, introduced by Howe, Leprevost and Poonen. By a new formulation, we link their results to Serre's assersion. Then, we recover a formula of Klein related to the question for complex threefolds. In this case the invariant is a modular form of weight 18, and the result is proved using theta functions identities.