This is a problem about the practicalities of removing singularities in multivariable complex functions.

In trying to derive the generating function (in two variables) for a certain problem in combinatorics, we have obtained an expression of the form $S(x,u) = N(x,u)/D(x,u)$ where $$ D(x,u) = (1 + u - u^2) x^2 + (3 u^3 - 2 u) x + u^2 (1 - u - u^2) $$ and $$ N(x,u) = P(x,u) + S(x) Q(x,u) + T(x) R(x,u) $$ Here $P$, $Q$ and $R$ are low degree polynomials in $x$ and $u$, and $S$ and $T$ are known algebraic functions (having convergent power series expansions in a neighbourhood of 0).

The theorem on singularity removal that we would hope to apply is that if $N/D$ is bounded on a neighbourhood of $(0,0)$ excluding the set where $D(x,u) = 0$, then the singularity which that set represents is actually removable, and there is an analytic $F(x,u) = N(x,u)/D(x,u)$ on a neighbourhood of $(0,0)$ wherever the right hand side is defined.

If it matters (probably not), note that setting $D(x,u) = 0$ and solving for $x$ actually gives $x$ as (one of two) rational functions of $u$ (since the discriminant is $5 u^6$). We can also verify (of course or this would be a very silly question indeed) that $N(x,u) = 0$ whenever $D(x,u) = 0$.

What's tripping us up is the practical issues of establishing the boundedness criteria. Can anyone make suggestions about how to do that?

For more specific details, the problem concerns the example discussed in section 4.1 of Generating permutations with restricted containers (pages 9 to 11) which, as it stands, is missing the final piece that an answer to this question would provide.