The Gliozzi's method in the Conformal Bootstrap consists in finding approximate solutions to the crossing equation $$ \sum_{\Delta,L} \mathsf{p}_{\Delta,L} \, \frac{v^{\Delta_\varphi}G_{\Delta,L}(u,v)-u^{\Delta_\varphi}G_{\Delta,L}(v,u)}{u^{\Delta_\varphi}-v^{\Delta_\varphi}}= 1\,.\tag{4} $$ where $u=\frac{x_{12}^2 x_{34}^2}{x_{13}^2 x_{24}^2},\,v = \frac{x_{14}^2 x_{23}^2}{x_{13}^2 x_{24}^2}$ are the crossing ratios of a four point function $\langle \phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\rangle$ of a scalar with conformal dimension $\Delta_\varphi$ in a generic CFT, $\mathsf{p}_{\Delta,l}$ are the squared OPE coefficients and $G_{\Delta,L}(u,v)$ are the conformal blocks.

Such approximate solutions are obtained by truncating the sum over $\Delta$ and $L$ to a finite number of terms. The crossing equation can be Taylor expanded around $u = v = 1/2$ thus yielding a single inhomogeneous equation $$ \sum_{\Delta,L} \mathsf{p}_{\Delta,L} \, \mathsf{f}^{(0,0)}_{\Delta_\varphi,\Delta_L} = 1\,,\tag{5} $$ and arbitrarily many homogeneous ones $$ \sum_{\Delta,L} \mathsf{p}_{\Delta,L} \, \mathsf{f}^{(2m,n)}_{\Delta_\varphi,\Delta_L} = 0\,,\quad (m,n\in \mathbb{N},\,m+n\neq 0)\,. \tag{6} $$ Here $\mathsf{f}^{(m,n)}_{\Delta_\varphi,\Delta_L}$ is the $m$th and $n$th derivative of the function defined in $(4)$ evaluated at $u=v=1/2$. Obviously $\mathsf{f}^{(0,0)}_{\Delta_\varphi,\Delta_L}$ is just the function evaluated at $u=v=1/2$. The precise definition is in the paper linked at the beginning but it's not important.

Now the author focuses only on $(6)$ and shows a condition for the existence of nontrivial solutions to that system. Consider the matrix ($i$ labels rows, $j$ columns) $$ \left(\mathsf{f}^{(m_i,n_i)}_{\Delta_\varphi,(\Delta_L)_j}\right)_{i,j}\,, $$ where $i$ runs through the various equations kept in $(6)$ and $j$ runs through the operators kept in the truncation. The condition for the existence of a nontrivial solution is that the above matrix must not have full rank. I agree with this statement but then he says that $(5)$ is simply a normalization condition, implying that we could always extend it to a solution of the full crossing equation $(5) + (6)$.

I believe that this is wrong if $(5)$ happens to be a linear combination of some of the equations in $(6)$. In this case indeed $(5)$ reduces to $0=1$. More formally, this is a simple application of Rouché–Capelli theorem: the solutions exists only if the matrices $$ \left(\begin{array}{c} \mathsf{f}^{(0,0)}_{\Delta_\varphi,(\Delta_L)_j}\\ \hline \mathsf{f}^{(m_i,n_i)}_{\Delta_\varphi,(\Delta_L)_j} \end{array} \right)\quad \mbox{and}\quad \left(\begin{array}{c|c} \mathsf{f}^{(0,0)}_{\Delta_\varphi,(\Delta_L)_j}& 1\\ \hline \mathsf{f}^{(m_i,n_i)}_{\Delta_\varphi,(\Delta_L)_j} & 0 \end{array} \right)\,, $$ have the same rank, and this cannot be if $\mathsf{f}^{(0,0)}_{\Delta_\varphi,(\Delta_L)_j}$ is in the span of $\mathsf{f}^{(m_i,n_i)}_{\Delta_\varphi,(\Delta_L)_j}$.

My question is: how can we make sure that this doesn't happen and thus that we can always extend a solution of the homogeneous system to a solution of the inhomogeneous one? Or, alternatively, I'm asking if there are examples where what I just described actually happens. I expect the solution, if it exists, to rely somehow on the properties of the conformal blocks.


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