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In the report by M. Singer (USA) read at Moscc)w University in October 1989 the following problem 'was formulated: for differential operator D = Pa ~z + Q a% to find such integer Ii, that any irreducible polynomial ~ dividing D$ has degree deg ~ < 1{. This problem arises when we want to find the symbolic solution of a differential equation ~ = R(z, y) where R is a raticmal function. It is proved in [1,2] that for any P and Q from C[Z, y] such K exists. Darboux gave partial result of this problem when deg P = deg Q = 1. In all other situations the solution of this problem was unknown. Here a solution when P and Q are homogeneous poly-nomials of equal degrees i.e. P(X, y) = ~mP(g)) Q(z, y) = z?''q(~) for some m, is proposed. Let the following equation is given Paf af @Q-=Af ay (1) P and Q are the given homogeneous polynomials of x and y of degree m and f, A are unknown polynomials of x and y. where Al and g are polynomials of x and z, p and q are polynomials of z only and deg p = deg q = m. Equation (1) has a form Np(%)g +z '-'(-zp(z) + q(z))~ = AI(z, z)g(:c, z) (2) Permission to copy without fee all or part of this material is granted providad that the copies are not made or distributed for direct commercial advantaga, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fse and/or specific permission. Let the greatest degree of the unknown x dividing the polynomial g be k. Then the left hand side of equation (2) can be divided by z rn+k-1 and to satisfy the equation it is necessary that Al be divided by Xm-i. On the other hand, degree in variable x of the left hand side of equation (2) is not greater than m + deg =q — 1 and degree of its right hand side equals deg .A1 + deg ,g, i.e. deg ZA1 < m – 1, that means Al = z '-la(z), and equation (2) may be written in the form ag (3) ZP(Z)~ + (–zp(z) + q(z))~ = a(~)9(x, Z) Let …

A bound of degree of irreducible eigenpolynomial of some differential operator