Equivariant compactifications of reductive groups

We study equivariant projective compactifications of reductive groups obtained by closing the image of a group in the space of operators of a projective representation. We describe the structure and the mutual position of their orbits under the action of the doubled group by left/right multiplications, the local structure in a neighborhood of a closed orbit, and obtain some conditions of normality and smoothness of a compactification. Our methods of research use the theory of equivariant embeddings of spherical homogeneous spaces and of reductive algebraic semigroups.Comment: 30 pages, AmSLaTeX. Bibliography: 36 item

Correspondences and singular varieties

What is generally known as the "Bloch--Srinivas method" consists of decomposing the diagonal of a smooth projective variety, and then considering the action of correspondences in cohomology. In this note, we observe that this same method can also be extended to singular and quasi--projective varieties. We give two applications of this observation: the first is a version of Mumford's theorem, the second is concerned with the Hodge conjecture for singular varieties.Comment: 11 pages. Comments welcome ! To appear in Monatsh. Math. (in slightly different version). arXiv admin note: text overlap with arXiv:1507.0448

A characterisation of Lie algebras via algebraic exponentiation

In this article we describe varieties of Lie algebras via algebraic exponentiation, a concept introduced by Gray in his Ph.D. thesis. For $\mathbb{K}$ an infinite field of characteristic different from $2$, we prove that the variety of Lie algebras over $\mathbb{K}$ is the only variety of non-associative $\mathbb{K}$-algebras which is a non-abelian locally algebraically cartesian closed (LACC) category. More generally, a variety of $n$-algebras $\mathcal{V}$ is a non-abelian (LACC) category if and only if $n=2$ and $\mathcal{V}=\mathsf{Lie}_\mathbb{K}$. In characteristic $2$ the situation is similar, but here we have to treat the identities $xx=0$ and $xy=-yx$ separately, since each of them gives rise to a variety of non-associative $\mathbb{K}$-algebras which is a non-abelian (LACC) category.Comment: The ancillary files contain the code used in the proofs. Final version to appear in Advances in Mathematic

Decomposing Jacobians of Curves over Finite Fields in the Absence of Algebraic Structure

We consider the issue of when the L-polynomial of one curve over \F_q divides the L-polynomial of another curve. We prove a theorem which shows that divisibility follows from a hypothesis that two curves have the same number of points over infinitely many extensions of a certain type, and one other assumption. We also present an application to a family of curves arising from a conjecture about exponential sums. We make our own conjecture about L-polynomials, and prove that this is equivalent to the exponential sums conjecture.Comment: 20 page

Computing Equilibria of Semi-algebraic Economies Using Triangular Decomposition and Real Solution Classification

In this paper, we are concerned with the problem of determining the existence of multiple equilibria in economic models. We propose a general and complete approach for identifying multiplicities of equilibria in semi-algebraic economies, which may be expressed as semi-algebraic systems. The approach is based on triangular decomposition and real solution classification, two powerful tools of algebraic computation. Its effectiveness is illustrated by two examples of application.Comment: 24 pages, 5 figure

Multiplicity Preserving Triangular Set Decomposition of Two Polynomials

In this paper, a multiplicity preserving triangular set decomposition algorithm is proposed for a system of two polynomials. The algorithm decomposes the variety defined by the polynomial system into unmixed components represented by triangular sets, which may have negative multiplicities. In the bivariate case, we give a complete algorithm to decompose the system into multiplicity preserving triangular sets with positive multiplicities. We also analyze the complexity of the algorithm in the bivariate case. We implement our algorithm and show the effectiveness of the method with extensive experiments.Comment: 18 page