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Divided differences of multivariate implicit functions
- BIT Num. Math
"... Abstract. Under general conditions, the equation g(x1,..., xq, y) = 0 implicitly defines y locally as a function of x1,..., xq. 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 in ..."
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Abstract. Under general conditions, the equation g(x1,..., xq, y) = 0 implicitly defines y locally as a function of x1,..., xq. 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. 1.
