The paper presents an experimental study investigating the impact of under-oxygenated conditions on the burning characteristics and flame extinction of horizontal transparent poly(methyl-methacrylate) (PMMA) slabs subjected to external heating. Small-sized solids are exposed to various constant levels of radiant heat flux in a nitrogen-diluted environment at ambient pressure. The study aims to understand the multivariable dependence of PMMA combustion on oxygen and heating levels. The analysis goes beyond the study of global and local variables to provide an understanding of the mechanisms involved, revealing a notable effect on burning characteristics and flame extinction mechanisms by blow-off. The results provide a detailed assessment of the surface energy balance and flame bulk properties. The results demonstrate good repeatability up to a certain oxygen limit, beyond which the flame combustion regime exhibits stochastic behaviour. The study of heat transfer mechanisms has revealed that flame radiation and external heating play a predominant role in the heating of horizontal solid fuels, In contrast, within the flame, convection is the main contributor to solid heating, rather than flame radiation. Solid characteristics show linear trends proportional to oxygen depletion, while gas phase characteristics show monotonic trends strongly influenced by the changing combustion regime. The surface temperature of the solid is highly sensitive to the test conditions that affect the energy balance. Dimensionless parameters are developed and data are correlated with existing literature. The analysis demonstrates that the linear or monotonic trends are maintained regardless of external heating. Finally, models are proposed and validated to predict both the extinction limit and the burning rate independently of irradiance and oxygen levels.