185 research outputs found
Four lectures on secant varieties
This paper is based on the first author's lectures at the 2012 University of
Regina Workshop "Connections Between Algebra and Geometry". Its aim is to
provide an introduction to the theory of higher secant varieties and their
applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in
Mathematics & Statistics), Springer/Birkhause
How to Integrate a Polynomial over a Simplex
This paper settles the computational complexity of the problem of integrating
a polynomial function f over a rational simplex. We prove that the problem is
NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin
and Straus. On the other hand, if the polynomial depends only on a fixed number
of variables, while its degree and the dimension of the simplex are allowed to
vary, we prove that integration can be done in polynomial time. As a
consequence, for polynomials of fixed total degree, there is a polynomial time
algorithm as well. We conclude the article with extensions to other polytopes,
discussion of other available methods and experimental results.Comment: Tables added with new experimental results. References adde
A new formula for the determinant of matrices
In this paper, we present a new formula for the determinant of a
matrix. We approach via the sparse optimization problem and derive the formula
through the Least Absolute Shrinkage and Selection Operator (LASSO). Our
formula has the potential to advance understanding of the algebraic structure
of determinants, such as an upper bound of the tensor rank, various notions to
measure complexity, and effective computational tools in exterior algebras. We
also address several numerical experiments which compare our formula with
built-in functions in a computer-algebra system.Comment: 11 pages, 1 figur
On real Waring decompositions of real binary forms
The Waring Problem over polynomial rings asks how to decompose a homogeneous
polynomial of degree as a finite sum of -{th} powers of linear
forms. In this work we give an algorithm to obtain a real Waring decomposition
of any given real binary form of length at most its degree. In fact, we
construct a semialgebraic family of Waring decompositions for . Some
examples are shown to highlight the difference between the real and the complex
case.Comment: 21 pages; typos correcte
Software for Exact Integration of Polynomials over Polyhedra
We are interested in the fast computation of the exact value of integrals of
polynomial functions over convex polyhedra. We present speed ups and extensions
of the algorithms presented in previous work. We present the new software
implementation and provide benchmark computations. The computation of integrals
of polynomials over polyhedral regions has many applications; here we
demonstrate our algorithmic tools solving a challenge from combinatorial voting
theory.Comment: Major updat
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