We apply the theory of algebraic polynomials to analytically study thetransonic properties of general relativistic hydrodynamic axisymmetricaccretion onto non-rotating astrophysical black holes. For such accretionphenomena, the conserved specific energy of the flow, which turns out to be oneof the two first integrals of motion in the system studied, can be expressed asa 8$^{th}$ degree polynomial of the critical point of the flow configuration.We then construct the corresponding Sturm's chain algorithm to calculate thenumber of real roots lying within the astrophysically relevant domain of$\mathbb{R}$. This allows, for the first time in literature, to {\itanalytically} find out the maximum number of physically acceptable solution anaccretion flow with certain geometric configuration, space-time metric, andequation of state can have, and thus to investigate its multi-criticalproperties {\it completely analytically}, for accretion flow in which thelocation of the critical points can not be computed without taking recourse tothe numerical scheme. This work can further be generalized to analyticallycalculate the maximal number of equilibrium points certain autonomous dynamicalsystem can have in general. We also demonstrate how the transition from amono-critical to multi-critical (or vice versa) flow configuration can berealized through the saddle-centre bifurcation phenomena using certaintechniques of the catastrophe theory.
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