Existence of Nontrivial Negative Resonances for Polynomial Ordinary Differential Equations With Painlevé Property

The Painlev\'e classification is one of the central problems in analytics theory of differential equations rooted in the XIX century. Although it saw many significant advances in analyzing certain classes of equations, the classification still remains an open problem especially for the higher-order equations. One of the main classical methods of Painlev\'e analysis is based on considering the resonance numbers corresponding to the possible indices of arbitrary coefficients in the Laurent expansion of the general solution in a neighborhood of a movable singularity. Complex and non-integer values of resonance numbers point out to existence of the movable critical singularities and positive integer numbers could be used to construct the said general solution. Also the equation always possesses at least one negative resonance number of $-1$ which corresponds to an arbitrary position of a movable pole. However our understanding of the role of nontrivial negative resonances different from $-1$ remains limited in spite of certain recent methodological advances related to it. And though in the lower-order classifications built so far such equations with nontrivial negative resonances have rather been a special case, the result of present work demonstrates that negative resonances are in fact common for the higher degree ordinary differential equations with Painlev\'e property. Specifically we'll prove that their presence is the necessary condition of the Painlev\'e property for the equations with degree of the leading terms higher than two.