A New Proof of the GGR Conjecture

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 $0\le j<k\le n$. A version of it for $n\ge 2$ 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.