On generators of bounded ratios of minors for totally positive matrices

We provide a method for factoring all bounded ratios of the form $$\det A(I_1\mathrm{\mid }{I}_1^{\mathrm{\prime }})\det A(I_2\mathrm{\mid }{I}_2^{\mathrm{\prime }})\mathrm{/}\det A(J_1\mathrm{\mid }{J}_1^{\mathrm{\prime }})\det A(J_2\mathrm{\mid }{J}_2^{\mathrm{\prime }})$$ where $A$ is a totally positive matrix, into a product of more elementary ratios each of which is bounded by 1, thus giving a new proof of Skandera’s result. The approach we use generalizes the one employed by Fallat et al. in their work on principal minors. We also obtain a new necessary condition for a ratio to be bounded for the case of non-principal minors.