arXiv subjects

Thiago Dias

Publications and source records attributed to Thiago Dias.

12 recordsLinked to original sources

arXiv2026

Brehm-Wintner-Conley Dimension, Pl\"ucker Coordinates, and Generalized Dziobek-Williams Equations for Central Configurations

We develop an algebraic framework for central configurations of the $n$-body problem with homogeneous potentials, grounded in the exterior algebra of the configuration space and the normalized shifted Brehm--Wintner--Conley (BWC) matrix $S$. Relating the kernel of $S$ to the Pl\"ucker coordinates of the configuration, we generalize the determinantal equations obtained by Williams (1938) for the planar five-body problem to central configurations of any dimension and any number of bodies, and derive the Dziobek--Williams equations $\det(S_I^J)=\kappa\, z_I z_J$, which exhibit the compound matrix $S^{(t)}$ as a rank-one matrix. Introducing the \emph{Brehm--Wintner--Conley dimension} $\operatorname{bwc}(x)=\operatorname{rank}(S)$ (an integer invariant that stratifies central configurations and measures vertical degeneracy), together with the Pl\"ucker--BWC coordinates attached to it, we prove that each stratum of central configurations with fixed dimension and Brehm--Wintner--Conley dimension admits a base-point-free map into a Veronese variety, factoring through a Grassmannian invariant; on the Dziobek stratum this recovers the Dziobek--Veronese geometry previously introduced by the author. We further describe universal determinantal relations satisfied by the minors of $S$ and expand explicitly the resulting systems for the planar five- and six-body problems. We also interpret the mass-weighted entries of $S$ as an equilibrium stress: under the MacMillan--Bartky sign condition a strictly convex central configuration underlies a cable--strut tensegrity, and a theorem of Connelly then forces $S$ to be negative semidefinite with nullity three, so that vertical degeneracy cannot occur in this regime.

math.DS↗

arXiv2026

Generalized Laura-Andoyer equations and the enumeration of some symmetrical classes of Dziobek configurations

We study the symmetrical Dziobek configurations where, in $\mathbb{R}^{d}$, there are $d$ bodies with unit masses at the vertices of a regular $(d-1)$-dimensional simplex of unit edge length and two more bodies with nonzero masses $s,k$ are on the line passing through the center of the simplex and being orthogonal to it. In the case of logarithmic potential, the finiteness is proved for all $s,k\neq 0, d>1$, and we obtain the bifurcation surface in the $(s,k,d)$-space through Gr\"obner basis computation. Using cylindrical algebraic decompositions, we find $197232$ sample points in the complement of the bifurcation surface. We propose a method to reduce the number to only $202$. By Hermite's root counting theorem, we find that, generically, there can be $0,1,2,3$ or $4$ concave, $1,2,3, $ or $4$ convex, and in totality, $1,2,3,4$ or $5$ such configurations for all dimensions $d>1$. For positive $s$ and $k$, generically, there is a unique convex configuration, while the number of concave ones can be $0,2$ or $4$. All possible combinations for the numbers described above are realized when $d=2$. We obtain a set of generalized Laura-Andoyer equations equivalent to the central configurations equations for all fixed number of bodies $n=d+h$ and configuration dimension $d$. For homogeneous force law with exponent $a\in \mathbb{R}$, we use the action of permutation group $S_d$ in the Laura-Andoyer equations to reduce the equivalent $\binom{d+2}{2}\binom{d}{2}$ Laura-Andoyer equations to only two generalized polynomial algebraic equations for the studied class of symmetric configurations with two variables representing the positions of the two bodies not at the vertices of the simplex in four parameters $a,d,s,k$.

math.DS↗

arXiv2026

The Veronese Geometry of Dziobek Configurations and Generic Finiteness for Homogeneous Potentials

The main contribution of this paper is the proof of the generic finiteness of Dziobek central configurations for a homogeneous potential and the derivation of a uniform upper bound for their number. By exploiting the isomorphism between the Veronese variety and the determinantal variety associated with the Dziobek conditions, we define the Dziobek-Veronese variety and apply the dimension of fibers theorem to analyze the projection from the space of configurations and masses to the space of masses. We prove that the fibers of this projection, representing the central configurations for a given mass vector, are finite for masses chosen outside a proper algebraic subvariety. Furthermore, we utilize that the Dziobek variety is defined by an intersection of quadrics to obtain a bound of Bezout type for the number of Dziobek configurations with fixed masses given by a power of $2$ with exponent quadratic in $n$. Unlike previous estimates tailored to specific potentials, this bound depends solely on the dimension $n$. For instance, for the four-body problem, our bound reduces to $8192$, which is lower than the bound of $8472$ established by Moeckel and Hampton. This suggests that the complexity of counting Dziobek configurations for generic masses is governed primarily by the ambient geometry of the configuration space, rather than by the non-linearity of the interaction potential.

math.DS↗

arXiv2025

The Inequity of Consumption-Based Tax Systems

This study examines the lack of redistributive effectiveness of consumption-based tax systems with respect to social fairness. Through numerical simulations, we explore the wealth exchanges among economic agents subject to flat consumption taxes, comparing universal redistribution with optimal targeted approaches. The results demonstrate that consumption taxes exhibit inherent regressivity, disproportionately burdening the poorest 40% of households who contribute over half of total tax revenue for most tax rates. The findings challenge the equity of consumption taxes and provide quantitative insights for designing more fair fiscal policies.

physics.soc-ph↗

arXiv2025

Economic Inequality between Groups in an a priori Stratified Society

We present an agent-based model of economic exchange in a society composed of two groups, representing two social groups and with different internal protection rules for the poor agents. The goal is to address the emerging wealth distribution when economic rules are not the same for all individuals. Individuals exchange wealth in pairwise interactions with no underlying lattice. The wealth, risk aversion factor, and group of the agents characterize their state. The wealth exchanged between two agents obeys a fair rule: the quantities put at stake by them are the same regardless of who wins. One agent can interact with another agent in the same or the other group, controlled by a rate which is a parameter of the model. Inter-group exchanges obey an exclusive protection rule, which can be understood as a public policy to reduce inequality. We show that the most protected group accumulates more wealth, has less inequality, and has higher mobility than the other group. The results of simulations are compared with income distribution in Brazil discriminated by race as an example of the application our model.

physics.soc-ph↗

arXiv2025

Homophilic Effects on Economic Inequality: A Dynamic Network Agent-Based Model

Wealth transactions are central to economic activity, and their particularities shape macroeconomic outcomes. We propose an agent-based model to investigate how homophily influences economic inequality. The model simulates wealth exchanges in a dynamic network composed of two groups, $A$ and $B$, differentiated by a homophily parameter $\delta$, which increases intragroup connections within $A$. Economic interactions alternate between conservative wealth exchanges and connection rewiring, both influenced by agents' wealth and $\delta$. We examine economic and network dynamics under varying levels of social protection $f$, which favor poorer agents in transactions. At low $f$, results reveal high inequality and link concentration, with $\delta$ impacting only transient dynamics. At high $f$, homophily becomes an economic advantage, as increasing $\delta$ directs wealth flow to group $A$. However, since this flow benefits the wealthiest agents, it simultaneously exacerbates internal inequality within the group. These findings show that homophily is a significant driver of inequality, directing wealth towards the homophilous group and worsening internal disparities.

physics.soc-ph↗

arXiv2023

Effectiveness of wealth-based vs exchange-based tax systems in reducing inequality

In the so-called ``fair'' models of peer-to-peer wealth exchanges, economic inequality tends to reach its maximum value asymptotically. This global trend is evident as the richest continuously accumulate a larger share of wealth at the expense of others. To address the mounting issue of inequality, different strategies of taxes and redistribution are commonly employed. Our study delves into the interplay between wealth and trade (consumption) tax bases, probing their impact on wealth distribution within wealth-conservative economies. The ultimate aim is to unearth an optimal framework that adeptly curbs inequality.Through a meticulous analysis of varying tax rates and the allocation of the collected tax to the most economically vulnerable strata, we unveil a compelling pattern resembling two distinct phases. These phases delineate the most effective systems for inequality mitigation. Our findings underscore the synergistic potential of amalgamating these tax systems, surpassing the individual efficacy of each. This synthesis beckons policymakers to weave together tax rates and precision-targeted redistribution, crafting tax systems that wield the potential for tangible and substantial reductions in economic disparity.

physics.soc-ph↗

arXiv2021

Counting square free monomial cremona maps

We give a complete list of square-free Cremona maps with at most six variables, up to equivalence classes. We also build an algorithm to count monomial square-free Cremona transformations. Using this algorithm, we obtain a complete list of monomial square-free Cremona transformations in seven variables.

math.AG↗

arXiv2020

Waring problems and the Lefschetz properties

We study three variations of the Waring problem for polynomials, concerning the Waring rank, the border rank and the cactus rank of a form and we show how the Lefschetz properties of the associated algebra affect them. The main tool is the theory of mixed Hessian matrix. We construct new families of wild forms, that is, forms whose cactus rank, of schematic nature, is bigger then the border rank, defined geometrically.

math.AC↗

arXiv2018

Generic finiteness for a class of symmetric planar central configurations of the six-body problem and the six-vortex problem

A symmetric planar central configuration of the Newtonian six-body problem $x$ is called cross central configuration if there are precisely four bodies on a symmetry line of $x$. We use complex algebraic geometry and Groebner basis theory to prove that for a generic choice of positive real masses $m_1,m_2,m_3,m_4,m_5=m_6$ there is a finite number of cross central configurations. We also show one explicit example of a configuration in this class. A part of our approach is based on relaxing the output of the Groebner basis computations. This procedure allows us to obtain upper bounds for the dimension of an algebraic variety. We get the same results considering cross central configurations of the six-vortex problem.

math.DS↗

arXiv2016

New equations for central configurations and generic finiteness

We consider the finiteness problem for central configurations of the $n-$body problem. We prove that, for $n\geq4$, there exists a (Zariski) closed subset $B$ in the mass space $\mathbb{R}^{n}$, such that if $(m_1,...,m_n) \in \mathbb{R}^n\setminus B$, then there is a finite number of corresponding classes of $(n-2)-$dimensional central configurations for potential associated to a semi-integer exponent. Also, we obtain trilinear homogeneous polynomial equations of degree $3$ for central configurations of fixed dimension and, for each integer $k \geq 1$, we show that the set of mutual distances associated to a $k-$dimensional central configuration is contained in a determinantal algebraic set.

math.DS↗

arXiv2015

Counting Square free Cremona monomial maps

We use combinatorics tools to reobtain the classification of monomial quadratic Cremona transformations in any number of variables given in \cite{SV2} and to classify and count square free cubic Cremona maps with at most six variables, up to isomorphism.

math.AC↗