17,929 research outputs found
A Polyhedral Homotopy Algorithm For Real Zeros
We design a homotopy continuation algorithm, that is based on numerically
tracking Viro's patchworking method, for finding real zeros of sparse
polynomial systems. The algorithm is targeted for polynomial systems with
coefficients satisfying certain concavity conditions. It operates entirely over
the real numbers and tracks the optimal number of solution paths. In more
technical terms; we design an algorithm that correctly counts and finds the
real zeros of polynomial systems that are located in the unbounded components
of the complement of the underlying A-discriminant amoeba.Comment: some cosmetic changes are done and a couple of typos are fixed to
improve readability, mathematical contents remain unchange
A Special Homotopy Continuation Method For A Class of Polynomial Systems
A special homotopy continuation method, as a combination of the polyhedral
homotopy and the linear product homotopy, is proposed for computing all the
isolated solutions to a special class of polynomial systems. The root number
bound of this method is between the total degree bound and the mixed volume
bound and can be easily computed. The new algorithm has been implemented as a
program called LPH using C++. Our experiments show its efficiency compared to
the polyhedral or other homotopies on such systems. As an application, the
algorithm can be used to find witness points on each connected component of a
real variety
A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time
We describe a deterministic algorithm that computes an approximate root of n
complex polynomial equations in n unknowns in average polynomial time with
respect to the size of the input, in the Blum-Shub-Smale model with square
root. It rests upon a derandomization of an algorithm of Beltr\'an and Pardo
and gives a deterministic affirmative answer to Smale's 17th problem. The main
idea is to make use of the randomness contained in the input itself
Rigid continuation paths I. Quasilinear average complexity for solving polynomial systems
How many operations do we need on the average to compute an approximate root
of a random Gaussian polynomial system? Beyond Smale's 17th problem that asked
whether a polynomial bound is possible, we prove a quasi-optimal bound
. This improves upon the previously known
bound.
The new algorithm relies on numerical continuation along \emph{rigid
continuation paths}. The central idea is to consider rigid motions of the
equations rather than line segments in the linear space of all polynomial
systems. This leads to a better average condition number and allows for bigger
steps. We show that on the average, we can compute one approximate root of a
random Gaussian polynomial system of~ equations of degree at most in
homogeneous variables with continuation steps. This is a
decisive improvement over previous bounds that prove no better than
continuation steps on the average
Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving
We derive efficient algorithms for coarse approximation of algebraic
hypersurfaces, useful for estimating the distance between an input polynomial
zero set and a given query point. Our methods work best on sparse polynomials
of high degree (in any number of variables) but are nevertheless completely
general. The underlying ideas, which we take the time to describe in an
elementary way, come from tropical geometry. We thus reduce a hard algebraic
problem to high-precision linear optimization, proving new upper and lower
complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding
Minimizing Higgs Potentials via Numerical Polynomial Homotopy Continuation
The study of models with extended Higgs sectors requires to minimize the
corresponding Higgs potentials, which is in general very difficult. Here, we
apply a recently developed method, called numerical polynomial homotopy
continuation (NPHC), which guarantees to find all the stationary points of the
Higgs potentials with polynomial-like nonlinearity. The detection of all
stationary points reveals the structure of the potential with maxima,
metastable minima, saddle points besides the global minimum. We apply the NPHC
method to the most general Higgs potential having two complex Higgs-boson
doublets and up to five real Higgs-boson singlets. Moreover the method is
applicable to even more involved potentials. Hence the NPHC method allows to go
far beyond the limits of the Gr\"obner basis approach.Comment: 9 pages, 4 figure
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