162 research outputs found
A Polyhedral Method to Compute All Affine Solution Sets of Sparse Polynomial Systems
To compute solutions of sparse polynomial systems efficiently we have to
exploit the structure of their Newton polytopes. While the application of
polyhedral methods naturally excludes solutions with zero components, an
irreducible decomposition of a variety is typically understood in affine space,
including also those components with zero coordinates. We present a polyhedral
method to compute all affine solution sets of a polynomial system. The method
enumerates all factors contributing to a generalized permanent. Toric solution
sets are recovered as a special case of this enumeration. For sparse systems as
adjacent 2-by-2 minors our methods scale much better than the techniques from
numerical algebraic geometry
Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories
The interplay rich between algebraic geometry and string and gauge theories
has recently been immensely aided by advances in computational algebra.
However, these symbolic (Gr\"{o}bner) methods are severely limited by
algorithmic issues such as exponential space complexity and being highly
sequential. In this paper, we introduce a novel paradigm of numerical algebraic
geometry which in a plethora of situations overcomes these short-comings. Its
so-called 'embarrassing parallelizability' allows us to solve many problems and
extract physical information which elude the symbolic methods. We describe the
method and then use it to solve various problems arising from physics which
could not be otherwise solved.Comment: 36 page
Global Structure of Curves from Generalized Unitarity Cut of Three-loop Diagrams
This paper studies the global structure of algebraic curves defined by
generalized unitarity cut of four-dimensional three-loop diagrams with eleven
propagators. The global structure is a topological invariant that is
characterized by the geometric genus of the algebraic curve. We use the
Riemann-Hurwitz formula to compute the geometric genus of algebraic curves with
the help of techniques involving convex hull polytopes and numerical algebraic
geometry. Some interesting properties of genus for arbitrary loop orders are
also explored where computing the genus serves as an initial step for integral
or integrand reduction of three-loop amplitudes via an algebraic geometric
approach.Comment: 35pages, 10 figures, version appeared in JHE
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