POWER GEOMETRY IN LOCAL RESOLUTION OF SINGULARITIES OF AN ALGEBRAIC CURVE

Keywords: power geometry, space curve, singular point, Newton polyhedron, normal cone, power transformations

Abstract

The main goal of this work is to provide a consistent set of general-purpose algorithms for analyzing singularities applicable to all types of equations. We present the main ideas and algorithms of power geometry and give an overview of some of its applications. We also present a procedure that allows us to distinguish all branches of a spatial curve near a singular point and calculate the parametric appearance of these branches with any degree of accuracy. For a specific case, we show how this algorithm works.

References

A.D. Bruno, Power Geometry in Algebraic and Differential Equations, orthHolland Mathematical Library, N, V.57, Elsevier, 2000.

A.D. Bruno, A.S. Soleev, Local uniformization of branches of a space curve and Newton polyhedra. Algebra and Analiz, 1991. Vol. 3, no. 1. P. 67-102.

A.S. Soleev, Algorithm of local resolution of singularities of a space curve, LNCS 3718, pp.405-415. Springer-Verlag, 2005.

A.S. Soleev, N.A.Soleeva Power Geometry for Finding Periodic Solutions in One System of ODE. Malaysian Journal of Mathematical Sciences, No 2, 2014.

A.S. Soleev, Singling out branches of an algebraic curve and Newton polyhedra, Dokl. Akad. Nauk SSSR 268 (1983), 1305-1307; (R) = Soviet Math Dokl. 27 (1983) (E).

A.D. Bruno, A.B. Batkhin Asymptotic solution of an algebraic equation, DAN 440:3 (2011) 295-300 (R) = Doklady Mathematics 84:2 (2011) 634639 (E).

Soleev A. Complicated Bifurcations of Periodic Solutions in some System of ODE. Canadian Mathem. Bulletin. Vol.39(3), 1996.p.360-366.

Views:

122

Downloads:

115

Published
2020-06-30
Citations
How to Cite
Akhmadjon Soleev. (2020). POWER GEOMETRY IN LOCAL RESOLUTION OF SINGULARITIES OF AN ALGEBRAIC CURVE. World Science, 1(6(58), 21-26. https://doi.org/10.31435/rsglobal_ws/30062020/7100
Section
Physics and Mathematics
Share