Complexity of Sparse Polynomial Solving: Homotopy on Toric Varieties and the Condition Metric
This paper investigates the cost of solving systems of sparse polynomial equations by homotopy continuation. First, a space of systems of n -variate polynomial equations is specified through n monomial bases. The natural locus for the roots of those systems is known to be a certain toric variety. Th...
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Veröffentlicht in: | Foundations of computational mathematics 2019-02, Vol.19 (1), p.1-53 |
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Zusammenfassung: | This paper investigates the cost of solving systems of sparse polynomial equations by homotopy continuation. First, a space of systems of
n
-variate polynomial equations is specified through
n
monomial bases. The natural locus for the roots of those systems is known to be a certain toric variety. This variety is a compactification of
(
C
\
{
0
}
)
n
, dependent on the monomial bases. A toric Newton operator is defined on that toric variety. Smale’s alpha theory is generalized to provide criteria of quadratic convergence. Two condition numbers are defined, and a higher derivative estimate is obtained in this setting. The Newton operator and related condition numbers turn out to be invariant through a group action related to the momentum map. A homotopy algorithm is given and is proved to terminate after a number of Newton steps which is linear on the condition length of the lifted homotopy path. This generalizes a result from Shub (Found Comput Math 9(2):171–178,
2009
.
https://doi.org/10.1007/s10208-007-9017-6
). |
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ISSN: | 1615-3375 1615-3383 |
DOI: | 10.1007/s10208-018-9375-2 |