Divided Differences of Multivariate Implicit Functions

TL;DR

This paper generalizes formulas for divided differences of implicitly defined multivariate functions, expressing them in terms of divided differences of the defining function and connecting combinatorial structures to derivatives.

Contribution

It extends recent univariate formulas to multivariate cases, involving polygonal partitions and plane trees, and derives derivative formulas from these differences.

Findings

Derived a formula for divided differences of multivariate implicit functions.

Connected combinatorial structures to derivatives of implicit functions.

Provided a generalization of existing univariate formulas.

Abstract

Under general conditions, the equation $g(x^1, ..., x^q, y) = 0$ implicitly defines $y$ locally as a function of $x^1, ..., x^q$. In this article, we express divided differences of $y$ in terms of divided differences of $g$, generalizing a recent formula for the case where $y$ is univariate. The formula involves a sum over a combinatorial structure whose elements can be viewed either as polygonal partitions or as plane trees. Through this connection we prove as a corollary a formula for derivatives of $y$ in terms of derivatives of $g$.