4,516 research outputs found
Mixed Discriminants
The mixed discriminant of n Laurent polynomials in n variables is the
irreducible polynomial in the coefficients which vanishes whenever two of the
roots coincide. The Cayley trick expresses the mixed discriminant as an
A-discriminant. We show that the degree of the mixed discriminant is a
piecewise linear function in the Plucker coordinates of a mixed Grassmannian.
An explicit degree formula is given for the case of plane curves.Comment: 17 page
An efficient tree decomposition method for permanents and mixed discriminants
We present an efficient algorithm to compute permanents, mixed discriminants and hyperdeterminants of structured matrices and multidimensional arrays (tensors). We describe the sparsity structure of an array in terms of a graph, and we assume that its treewidth, denoted as ω, is small. Our algorithm requires Õ(n2[superscript ω]) arithmetic operations to compute permanents, and Õ(n[superscript 2] + n3[superscript ω]) for mixed discriminants and hyperdeterminants. We finally show that mixed volume computation continues to be hard under bounded treewidth assumptions. Keywords: Permanent; Structured array; Mixed discriminant;
Treewidth; HyperdeterminantUnited States. Air Force Office of Scientific Research (Grant FA9550-11-1-0305
The Alexandrov-Fenchel type inequalities, revisited
Various Alexandrov-Fenchel type inequalities have appeared and played
important roles in convex geometry, matrix theory and complex algebraic
geometry. It has been noticed for some time that they share some striking
analogies and have intimate relationships. The purpose of this article is to
shed new light on this by comparatively investigating them in several aspects.
\emph{The principal result} in this article is a complete solution to the
equality characterization problem of various Alexandrov-Fenchel type
inequalities for intersection numbers of nef and big classes on compact
K\"{a}hler manifolds, extending earlier results of Boucksom-Favre-Jonsson,
Fu-Xiao and Xiao-Lehmann. Our proof combines a result of Dinh-Nguy\^{e}n on
K\"{a}hler geometry and an idea in convex geometry tracing back to Shephard. In
addition to this central result, we also give a geometric proof of the complex
version of the Alexandrov-Fenchel type inequality for mixed discriminants and a
determinantal type generalization of various Alexandrov-Fenchel type
inequalities.Comment: 18 pages, slightly revised version stressing our principal result,
comments welcom
A generalized Lieb's theorem and its applications to spectrum estimates for a sum of random matrices
In this paper we prove the concavity of the -trace functions, , on the convex cone of all positive
definite matrices. denotes the elementary
symmetric polynomial of the eigenvalues of . As an application, we use the
concavity of these -trace functions to derive tail bounds and expectation
estimates on the sum of the largest (or smallest) eigenvalues of a sum of
random matrices.Comment: 22 page
Plane mixed discriminants and toric jacobians
Polynomial algebra offers a standard approach to handle several problems in
geometric modeling. A key tool is the discriminant of a univariate polynomial,
or of a well-constrained system of polynomial equations, which expresses the
existence of a multiple root. We concentrate on bivariate polynomials and
establish an original formula that relates the mixed discriminant of two
bivariate Laurent polynomials with fixed support, with the sparse resultant of
these polynomials and their toric Jacobian. This allows us to obtain a new
proof for the bidegree of the mixed discriminant as well as to establish
multipicativity formulas arising when one polynomial can be factored.Comment: 16 page
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