Combinatorial and Gaussian Foundations of Rational Nth Root Approximations: Theorems and Conjectures

TL;DR

This paper introduces the biroot method for rational nth root approximation, leveraging combinatorial and Gaussian structures, with new conjectures, proofs for special cases, and evidence of superior convergence over traditional methods.

Contribution

It proposes a novel biroot approach based on binomial and Gaussian structures, formulates three main conjectures, and demonstrates improved convergence compared to existing approximation techniques.

Findings

Proved the square root case of the Binomial Biroot Conjecture.

Formulated three main conjectures on root approximation structures.

Showed computational evidence of superior convergence over Taylor and Pade methods.

Abstract

We present an approach (the biroot method) for nth root approximation that yields closed-form rational functions with coefficients derived from binomial structures, Gaussian functions, or qualifying DAG structures. The method emerges from an analysis of Newton's method applied to root extraction, revealing that successive iterations generate coefficients following rows of Pascal's triangle in an alternating numerator-denominator pattern. After further exploration of these patterns, we formulate three main conjectures: (1) the Binomial Biroot Conjecture establishing the fundamental alternating coefficient structure to approximate nth roots (for which we prove the square root case and optimal parameter conditions), (2) the Gaussian Biroot Conjecture, and (3) the General DAG Biroot Conjecture showing a structural invariance to nth root approximation using arbitrary linearly-constructed directed acyclic graphs (DAGs). Computational evidence demonstrates superior convergence properties when compared to Taylor series and Pad\'e approximations, especially considering the more direct and less computationally intensive approach to the biroot function construction. A computational framework has been developed to support systematic exploration of the biroot method's parameter space and to enable extensive numerical and symbolic analysis. The method provides both theoretical insights and computational significance by connecting combinatorial structures to nth root rational approximation theory.