# Existence of solutions to crossing with the Gliozzi's method

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.