A New Proof of the GGR Conjecture

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 $n(n+1)\mathrm{/}2$ generalized Riemann derivatives, $$D_{k,-j}f(x)=\underset{h\to 0}{\lim }\frac{1}{h^k}\sum _{i=0}^k(-1{)}^i\left(\genfrac{}{}{0px}{}{k}{i}\right)f(x+(k-i-j)h),$$ 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.