I recently constructed a proof that a computable universal function exists for the class of polynomials of $n$-variables. To this end, I adopted the graded lexicographical monomial order. However, I don't have a proof that the $k$-th term in this order is a recursive function. Can I get some help?
Attempt: If $x_{(k)}=x^a$ is the $k$-th monomial then $x_{(k+1)}=x_{(k)}x_n$ or $x_{(k+1)}=\frac{x_{(k)}x_i}{x_j}$, depending on the multi-index $a$, but I can't get the cases right.