4,086 research outputs found
On semidefinite representations of plane quartics
This note focuses on the problem of representing convex sets as projections
of the cone of positive semidefinite matrices, in the particular case of sets
generated by bivariate polynomials of degree four. Conditions are given for the
convex hull of a plane quartic to be exactly semidefinite representable with at
most 12 lifting variables. If the quartic is rationally parametrizable, an
exact semidefinite representation with 2 lifting variables can be obtained.
Various numerical examples illustrate the techniques and suggest further
research directions
Interpolation with bilinear differential forms
We present a recursive algorithm for modeling with bilinear differential forms. We discuss applications of this algorithm for interpolation with symmetric bivariate polynomials, and for computing storage functions for autonomous systems
Algebraic statistical models
Many statistical models are algebraic in that they are defined in terms of
polynomial constraints, or in terms of polynomial or rational parametrizations.
The parameter spaces of such models are typically semi-algebraic subsets of the
parameter space of a reference model with nice properties, such as for example
a regular exponential family. This observation leads to the definition of an
`algebraic exponential family'. This new definition provides a unified
framework for the study of statistical models with algebraic structure. In this
paper we review the ingredients to this definition and illustrate in examples
how computational algebraic geometry can be used to solve problems arising in
statistical inference in algebraic models
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