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Rota-Baxter operators present a natural generalisation of integration byparts formula for the integral operator. In 2015, Zheng, Guo, and Rosenkranzconjectured that every injective Rota-Baxter operator of weight zero on thepolynomial algebra $\mathbb{R}[x]$ is a composition of the multiplication by anonzero polynomial and a formal integration at some point. We confirm thisconjecture over any field of characteristic zero. Moreover, we establish astructure of an ind-variety on the moduli space of these operators and describean additive structure of generic modality two on it. Finally, we provide aninfinitely transitive action on codimension one subsets.

Injective Rota-Baxter operators of weight zero on $F[x]$