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The Painlev\'e classification is one of the central problems in analyticstheory of differential equations rooted in the XIX century. Although it sawmany significant advances in analyzing certain classes of equations, theclassification still remains an open problem especially for the higher-orderequations. One of the main classical methods of Painlev\'e analysis is based onconsidering the resonance numbers corresponding to the possible indices ofarbitrary coefficients in the Laurent expansion of the general solution in aneighborhood of a movable singularity. Complex and non-integer values ofresonance numbers point out to existence of the movable critical singularitiesand positive integer numbers could be used to construct the said generalsolution. Also the equation always possesses at least one negative resonancenumber of $-1$ which corresponds to an arbitrary position of a movable pole.However our understanding of the role of nontrivial negative resonancesdifferent from $-1$ remains limited in spite of certain recent methodologicaladvances related to it. And though in the lower-order classifications built sofar such equations with nontrivial negative resonances have rather been aspecial case, the result of present work demonstrates that negative resonancesare in fact common for the higher degree ordinary differential equations withPainlev\'e property. Specifically we'll prove that their presence is thenecessary condition of the Painlev\'e property for the equations with degree ofthe leading terms higher than two.

Existence of Nontrivial Negative Resonances for Polynomial Ordinary Differential Equations With Painlev\'e Property