A Gr\"obner--Shirshov Basis for Nilpotent Rota--Baxter Algebras of Weight Zero
We construct an explicit Gr\"obner--Shirshov basis for free associative Rota--Baxter algebras of weight zero with nilpotent operator $R^n=0$, where $n\ge 2$. First, we define a monomial order on the standard linear basis $RS(X)$ of the free algebra $R\mathrm{As}\langle X\rangle$ and establish fundamental identities for Rota--Baxter operators. For the case $n=2$, the basis consists of the Rota--Baxter relation $R(u)R(v)\to R(uR(v))+R(R(u)v)$ and the nilpotency relation $R(R(w))\to 0$. For general $n\ge 3$, we prove that the Gr\"obner--Shirshov basis is finite and consists of six families of relations $(R1)$--$(R6)$ derived from resolving all composition ambiguities. Using the Composition-Diamond Lemma, we describe the corresponding irreducible basis $\operatorname{Irr}(S)$, which provides normal forms for elements in the quotient algebra. This result gives a complete solution to the word problem for nilpotent Rota--Baxter algebras and establishes their operadic Gr\"obner--Shirshov basis.