SOLUTIONS OF THE 2ND AND 4TH PAINLEVE EQUATIONS, I

A rigorous proof of the irreducibility of the second and fourth Painle ve equations is given by applying Umemura's theory on algebraic differ ential equations ([26], [27], [28]) to the two equations. The proof co nsists of two parts: to determine a necessary condition for the parame ters of the existence of principal ideals invariant under the Hamilton ian vector field; to determine the principal invariant ideals for a pa rameter where the principal invariant ideals exist. Our method is rele ased from complicated calculation, and applicable to the proof of the irreducibility of the third, fifth and sixth equation (e.g. [32]).