Finite systems of equations and implicit functions

TL;DR

This paper examines the classical Ruckert-Lefschetz scheme for implicit functions defined by finite systems of equations, demonstrating its limitations and proposing modifications for finite coefficient calculations.

Contribution

It identifies the limitations of the classical scheme and introduces modifications enabling analysis with finite Taylor coefficients.

Findings

Classical scheme is not generally applicable.

Proposed modifications facilitate finite coefficient calculations.

Enhances computational approaches to implicit functions.

Abstract

The classical Ruckert-Lefschetz scheme of analysis of implicit functions (defined by finite systems of n analytical equations with n unknowns) is studied from the point of view of calculations with finite number coefficients in Taylor expansions for left hand parts of corresponding equations. It is proved that this scheme is not applicable in the general case. It is offered some modifications allowing to lead the analysis of implicit functions to the calculation with finite number of coefficients.