A New Proof of the GGR Conjecture

TL;DR

This paper presents a simple, inductive algebraic proof of the GGR conjecture, establishing the equivalence between the existence of Peano derivatives and generalized Riemann derivatives for all positive integers n.

Contribution

The authors provide a new, straightforward algebraic proof of the GGR conjecture, simplifying previous complex combinatorial approaches.

Findings

Proved the GGR conjecture using algebraic methods.

Established the equivalence between Peano derivatives and generalized Riemann derivatives.

Simplified the proof process compared to previous non-inductive methods.

Abstract

For each positive integer $n$, function $f$, and point $x$, the 1998 conjecture by Ghinchev, Guerragio, and Rocca states that the existence of the $n$-th Peano derivative $f_{(n)}(x)$ is equivalent to the existence of all $n(n+1)/2$ generalized Riemann derivatives, \[ D_{k,-j}f(x)=\lim_{h\rightarrow 0}\frac 1{h^{k}}\sum_{i=0}^k(-1)^i\binom{k}{i}f(x+(k-i-j)h), \] for $j,k$ with $0\leq j<k\leq n$. A version of it for $n\geq 2$ replaces all $-j$ with $j$ and eliminates all $j=k-1$. Both the GGR conjecture and its version were recently proved by the authors using non-inductive proofs based on highly non-trivial combinatorial algorithms. This article provides a simple, inductive, algebraic proof of each of these theorems, based on a reduction to (Laurent) polynomials.