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

TL;DR

This work introduces the biroot method for closed-form nth root rational approximations, deriving coefficient patterns from Pascal's triangle, Gaussian sampling, and generalized DAGs. It establishes a square-root convergence proof for the Binomial Biroot, proposes a central centering parameter $c$, and generalizes to arbitrary $n$ with computational evidence supporting convergence and conjectured invariances. A Gaussian Biroot variant leverages the Central Limit Theorem to achieve markedly faster convergence, while a DAG Biroot extends the framework to linearly constructed graphs with observed invariances. Together, these results connect combinatorial structures to practical root-approximation theory and offer a computational framework for exploring parameter spaces, error analysis, and potential hardware implementations. The work also identifies three main conjectures—Binomial Biroot, Gaussian Biroot, and DAG Biroot—as focal points for rigorous future development and theoretical grounding.

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 Padé 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.

Figures (16)

• Figure 1: Pascal's triangle with highlighted rows. It should be noted that each recursive step $k$ takes us to the $2^{k+1}$th row of Pascal's triangle.
• Figure 2: Red values are the coefficients of the numerator and Green values are the coefficients of denominator. Notice that coefficients are in pairs of two and are being sampled at every third position in the row.
• Figure 3: $c=1$: Convergence of $x$ in the interval $[0, 10^4]$ when varying $m$ over $[n+1, 200]$, and $n$ over $3, 4, 5$, and $6$.
• Figure 4: Gaussian Biroot convergence analysis with $c=1$. Note the significantly faster convergence compared to Binomial Biroot results, achieved with lower maximum order ($m \leq 70$ versus $m \leq 200$).
• Figure 5: DAG Biroot convergence analysis with $c=1$ for the DAG with random basin $[5, 2, 7, 1, 8]$ and arbitrary node arity 3

Theorems & Definitions (17)

• Definition 1.1: Biroot Approximant
• Remark
• Conjecture 3.1
• Remark
• Lemma 3.2: Binomial Identity for Even and Odd Terms
• proof
• Theorem 3.3: Square Root Case
• proof
• Conjecture 3.4: Generalized Binomial Biroot
• Remark