61 research outputs found
Waring-like decompositions of polynomials - 1
Let be a homogeneous form of degree in variables. A Waring
decomposition of is a way to express as a sum of powers of
linear forms. In this paper we consider the decompositions of a form as a sum
of expressions, each of which is a fixed monomial evaluated at linear forms.Comment: 12 pages; Section 5 added in this versio
Codimension one decompositions and Chow varieties
A presentation of a degree form in variables as the sum of
homogenous elements ``essentially'' involving variables is called a {\em
codimension one decomposition}. Codimension one decompositions are introduced
and the related Waring Problem is stated and solved. Natural schemes describing
the codimension one decompositions of a generic form are defined. Dimension and
degree formulae for these schemes are derived when the number of summands is
the minimal one; in the zero dimensional case the scheme is showed to be
reduced. These results are obtained by studying the Chow variety
of zero dimensional degree cycles in \PP^n. In particular, an explicit
formula for is determined
Orbitopes
An orbitope is the convex hull of an orbit of a compact group acting linearly
on a vector space. These highly symmetric convex bodies lie at the crossroads
of several fields, in particular convex geometry, optimization, and algebraic
geometry. We present a self-contained theory of orbitopes, with particular
emphasis on instances arising from the groups SO(n) and O(n). These include
Schur-Horn orbitopes, tautological orbitopes, Caratheodory orbitopes, Veronese
orbitopes and Grassmann orbitopes. We study their face lattices, their
algebraic boundary hypersurfaces, and representations as spectrahedra or
projected spectrahedra.Comment: 37 pages. minor revisions of origina
- âŠ