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Polyhedral methods in numerical algebraic geometry. (English) Zbl 1181.65073

Bates, Daniel J. (ed.) et al., Interactions of classical and numerical algebraic geometry. A conference in honor of Andrew Sommese, Notre Dame, IN, USA, May 22–24, 2008. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4746-6/pbk). Contemporary Mathematics 496, 243-263 (2009).
Summary: In numerical algebraic geometry witness sets are numerical representations of positive dimensional solution sets of polynomial systems. Considering the asymptotics of witness sets we propose certificates for algebraic curves. These certificates are the leading terms of a Puiseux series expansion of the curve starting at infinity. The vector of powers of the first term in the series is a tropism.
For proper algebraic curves, we relate the computation of tropisms to the calculation of mixed volumes. With this relationship, the computation of tropisms and Puiseux series expansions could be used as a preprocessing stage prior to a more expensive witness set computation. Systems with few monomials have fewer isolated solutions and fewer data are needed to represent their positive dimensional solution sets.
For the entire collection see [Zbl 1175.14001].

MSC:

65H10 Numerical computation of solutions to systems of equations
65H04 Numerical computation of roots of polynomial equations
14Q05 Computational aspects of algebraic curves

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Kronecker; Gfan; PHoM
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