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First approximations of algebraic equations. (English. Russian original) Zbl 0886.32006

Russ. Acad. Sci., Dokl., Math. 49, No. 2, 291-293 (1994); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 335, No. 3, 277-278 (1994).
It is considered an absolutely convergent sum \[ f= \sum f_\Omega X^\Omega \quad \text{for} \quad \Omega \in {\mathcal D}, \] where \(X=(x_1, \dots, x_n) \in\mathbb{C}^n\), \(\Omega= (q_1, \dots, q_n) \in R^n_1\), \(X^\Omega =x_1^{q_1} \dots x_n^{q_n}\), the constant coefficients \(f_\Omega \in \mathbb{C}\), \({\mathcal D}\) is a some point set in \(R^n_1\). The closure of the convex hull of set \({\mathcal D}\) is called the Newton polyhedron of the sum \(f\).
The first approximations of the function \(f\) as \(X\to X^0\) are distinguished with using Newton polyhedrons.

MSC:

32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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