How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties

Let $\mathcal D_{d,k}$ denote the discriminant variety of degree $d$ polynomials in one variable with at least one of its roots being of multiplicity $\geq k$. We prove that the tangent cones to $\mathcal D_{d,k}$ span $\mathcal D_{d,k-1}$ thus, revealing an extreme ruled nature of these varieties. The combinatorics of the web of affine tangent spaces to $\mathcal D_{d,k}$ in $\mathcal D_{d,k-1}$ is directly linked to the root multiplicities of the relevant polynomials. In fact, solving a polynomial equation $P(z) = 0$ turns out to be equivalent to finding hyperplanes through a given point P(z)\in \mathcal D_{d,1} \approx \A^d which are tangent to the discriminant hypersurface $\mathcal D_{d,2}$. We also connect the geometry of the Vi\`{e}te map \mathcal V_d: \A^d_{root} \to \A^d_{coef}, given by the elementary symmetric polynomials, with the tangents to the discriminant varieties $\{\mathcal D_{d,k}\}$. Various $d$-partitions $\{\mu\}$ provide a refinement $\{\mathcal D_\mu^\circ\}$ of the stratification of \A^d_{coef} by the $\mathcal D_{d,k}$'s. Our main result, Theorem 7.1, describes an intricate relation between the divisibility of polynomials in one variable and the families of spaces tangent to various strata $\{\mathcal D_\mu^\circ\}$.Comment: 43 pages, 12 figure

A Statistical Model of Abstention under Compulsory Voting

Invalid voting and electoral absenteeism are two important sources of abstention in compulsory voting systems. Previous studies in this area have not considered the correlation between both variables and ignored the compositional nature of the data, potentially leading to unfeasible results and discarding helpful information from an inferential standpoint. In order to overcome these problems, this paper develops a statistical model that accounts for the compositional and hierarchical structure of the data and addresses robustness concerns raised by the use of small samples that are typical in the literature. The model is applied to analyze invalid voting and electoral absenteeism in Brazilian legislative elections between 1945 and 2006 via MCMC simulations. The results show considerable differences in the determinants of both forms of non-voting; while invalid voting was strongly positively related both to political protest and to the existence of important informational barriers to voting, the influence of these variables on absenteeism is less evident. Comparisons based on posterior simulations indicate that the model developed in this paper fits the dataset better than several alternative modeling approaches and leads to different substantive conclusions regarding the effect of different predictors on the both sources of abstention