High order vanishing theorems for non-simple blowup solutions of singular Liouville equation

For a singular Liouville equation, it is plausible that a non-simple blowup phenomenon occurs around a quantized singular pole. The presence of complex blowup profiles of bubbling solutions presents substantial challenges in applications. In this article, we demonstrate that under natural assumptions, non-simple blowup takes place only when the derivatives of certain coefficient functions approach zero. Our main result encompasses all previous findings and determines the vanishing order for any specific quantized singular source. Our theorems can be utilized not only to eliminate multiple non-simple blowup scenarios in applications but also to investigate blowup solutions with moving poles.