Bivariate systems of polynomial equations with roots of high multiplicity

Given a bivariate system of polynomial equations with fixed support sets [Formula: see text] it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity [Formula: see text] for all [Formula: see text] in the range [Formula: see text], where [Formula: see text] is the set of all integral vectors that shift B to a subset of [Formula: see text]. As an application, we classify all pairs [Formula: see text] such that the system supported at [Formula: see text] does not have a solution of multiplicity higher than [Formula: see text].

On μ-symmetric polynomials

We study functions of the roots of an integer polynomial [Formula: see text] with [Formula: see text] distinct roots [Formula: see text] of multiplicity [Formula: see text], [Formula: see text]. Traditionally, root functions are studied via the theory of symmetric polynomials; we generalize this theory to [Formula: see text]-symmetric polynomials. We initiate the study of the vector space of [Formula: see text]-symmetric polynomials of a given degree [Formula: see text] via the concepts of [Formula: see text]-gist and [Formula: see text]-ideal. In particular, we are interested in the root function [Formula: see text]. The D-plus discriminant of [Formula: see text] is [Formula: see text]. This quantity appears in the complexity analysis of the root clustering algorithm of Becker et al. (ISSAC 2016). We conjecture that [Formula: see text] is [Formula: see text]-symmetric, which implies [Formula: see text] is rational. To explore this conjecture experimentally, we introduce algorithms for checking if a given root function is [Formula: see text]-symmetric. We design three such algorithms: one based on Gröbner bases, another based on canonical bases and reduction, and the third based on solving linear equations. Each of these algorithms has variants that depend on the choice of a basis for the [Formula: see text]-symmetric functions. We implement these algorithms (and their variants) in Maple and experiments show that the latter two algorithms are significantly faster than the first.

Interplay of Coulomb and symmetry potential in peak fragment production in asymmetric collisions

Role of the nuclear symmetry potential and Coulomb potential is explored on the peak energy of intermediate mass fragments ([Formula: see text]) and on peak multiplicity ([Formula: see text]) and their dependence on mass asymmetry of the reaction is also investigated. The calculations are done using Isospin-dependent Quantum Molecular Dynamics (IQMD) model. We also showed that the momentum-dependent interactions have uniform effects of [Formula: see text] and these effects are independent of mass asymmetry of the reaction. Further, we see that isospin effects that enter through the Coulomb and symmetry potential show much significant role as one increases the mass asymmetry of reaction. Mass asymmetric reactions thus serve a sensitive tool to investigate the nuclear symmetry energy effects.

Computing subschemes of the border basis scheme

A good way of parameterizing zero-dimensional schemes in an affine space [Formula: see text] has been developed in the last 20 years using border basis schemes. Given a multiplicity [Formula: see text], they provide an open covering of the Hilbert scheme [Formula: see text] and can be described by easily computable quadratic equations. A natural question arises on how to determine loci which are contained in border basis schemes and whose rational points represent zero-dimensional [Formula: see text]-algebras sharing a given property. The main focus of this paper is on giving effective answers to this general problem. The properties considered here are the locally Gorenstein, strict Gorenstein, strict complete intersection, Cayley–Bacharach, and strict Cayley–Bacharach properties. The key characteristic of our approach is that we describe these loci by exhibiting explicit algorithms to compute their defining ideals. All results are illustrated by nontrivial, concrete examples.

Amenable covers and ℓ1-invisibility

Let [Formula: see text] be a topological space admitting an amenable cover of multiplicity [Formula: see text]. We show that, for every [Formula: see text] and every [Formula: see text], the image of [Formula: see text] in the [Formula: see text]-homology module [Formula: see text] vanishes. This strengthens previous results by Gromov and Ivanov, who proved, under the same assumptions, that the [Formula: see text]-seminorm of [Formula: see text] vanishes.

On a special case of Wilf’s conjecture

Let [Formula: see text] be a numerical semigroup with embedding dimension [Formula: see text], minimal set of generators [Formula: see text], Frobenius number [Formula: see text], multiplicity [Formula: see text] and genus [Formula: see text]. In this paper, we prove that Wilfs conjecture i.e. the inequality [Formula: see text] holds for [Formula: see text] when [Formula: see text] is a basis for [Formula: see text]

The Automorphic Discrete Spectrum of $\textrm{Mp}_4$

We prove the multiplicity formula for the automorphic discrete spectrum of the metaplectic group $\textrm{Mp}_4$ of rank $2$.

Analysis of Bose–Einstein correlation at 7 TeV by LHCb collaboration based on stochastic approach

The Bose–Einstein correlation (BEC) in forward region [Formula: see text] measured at 7 TeV in the Large Hadron Collider (LHC) by the LHCb collaboration is analyzed using two conventional formulas of different types named CF[Formula: see text] and CF[Formula: see text]. The first formula is well known and contains the degree of coherence [Formula: see text] and the exchange function [Formula: see text] from the BE statistics. The second formula is an extended formula (CF[Formula: see text]) that contains the second degree of coherence [Formula: see text] and the second exchange function [Formula: see text] in addition to CF[Formula: see text]. To examine the physical meaning of the parameters estimated by CF[Formula: see text], we analyze the LHCb BEC data by using a stochastic approach of the three-negative binomial distribution and the three-generalized Glauber–Lachs formula. Our results reveal that the BEC at 7 TeV consisted of three activity intervals defined by the multiplicity [Formula: see text] ([8, 18], [19, 35], and [36, 96]) can be well explained by CF[Formula: see text].