[Paper Review] Calculus of linear extensions and Newton interpolation

Gennadiy Georgievich Ilyuta|ArXiv.org|Nov 30, 2009

Advanced Numerical Analysis Techniques3 citations

TL;DR

This paper establishes a novel connection between Greene's sums over linear extensions of partially ordered sets (posets) and Newton's divided differences, enabling new interpolation formulas indexed by sequences of posets. By leveraging residue calculus and generating functions, it derives generalized Newton interpolation series and identities linking divided differences to Arnold's differential forms, with applications to rational function identities and multivariate interpolation.

ABSTRACT

We use Newton divided differences for calculation of Greene sums -- the rational functions determined by linear extensions of partially ordered sets. Identities for Greene sums generate relations for Newton divided differences and Arnold differential forms. Also generalizations of the Newton interpolation series which are indexed by sequences of partially ordered sets are received.

Motivation & Objective

• To establish a calculus of linear extensions for rational functions defined by posets.
• To derive new interpolation formulas indexed by sequences of posets using residue calculus.
• To generate identities for Newton divided differences and Arnold differential forms via Greene sum symmetries.
• To unify rational function identities and multivariate interpolation through poset decomposition.

Proposed method

• Use Newton divided differences to compute Greene sums over linear extensions of posets.
• Apply complex analysis and Cauchy's residue theorem to express analytic functions as integrals involving Greene sums.
• Decompose linear extensions via poset structure to derive identities for divided differences.
• Construct generalized Newton interpolation series indexed by sequences of posets.
• Utilize generating functions and lattice path models to interpret multivariate interpolation.
• Derive relations between divided differences and Arnold’s logarithmic differential forms via poset cycles.

Experimental results

Research questions

• RQ1 How can Greene sums over linear extensions of posets be expressed using Newton divided differences?
• RQ2 What new interpolation series arise from indexing Newton’s formula by sequences of posets?
• RQ3 How do identities in Greene sums generate relations for divided differences and Arnold differential forms?
• RQ4 Can multivariate interpolation be systematically derived from poset decompositions and residue integrals?
• RQ5 What is the role of poset cycles in generating functional identities for rational functions?

Key findings

• Greene sums generate identities for Newton divided differences, such as ΔXn[F] = ∑i F(xi)/f’(xi), linking them to residue integrals.
• The generalized Newton interpolation series is derived via residue integration, recovering Lagrange interpolation and Taylor series as special cases.
• For posets with tree-like Hassse diagrams, the interpolation formula simplifies to F(x) = (1/2πi)∫[(D(z)−D(x))/(z−x)]F(z)/D(z) dz + D(x)(1/2πi)∫F(z)/[(z−x)D(z)] dz.
• The method yields explicit formulas for F(x) when the poset Pn has a unique maximum z and n minimal elements x1,…,xn, giving L_F(x) = f(x)∑α∈Sn F(xα(1))/∏(x−xα(i))(xα(i)−xα(i+1)).
• Relations between divided differences and Arnold’s differential forms ωij = (1/2πi)d(xi−xj)/(xi−xj) are derived from non-linear 3-element posets.
• The paper generalizes classical identities such as the Grassmann-Plücker relation and Euler-Jacobi formula through poset-based decomposition of linear extensions.