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Black hole generalized p-brane solutions for a wide class of intersectionrules are obtained. The solutions are defined on a manifold that contains aproduct of n - 1 Ricci-flat internal spaces. They are defined up to a set offunctions H_s obeying non-linear differential equations equivalent to Toda-typeequations with certain boundary conditions imposed. A conjecture on polynomialstructure of governing functions H_s for intersections related to semisimpleLie algebras is suggested. This conjecture is proved for Lie algebras: A_m,C_{m+1}, m > 0. For simple Lie algebras the powers of polynomials coincide withthe components of twice the dual Weyl vector in the basis of simple coroots.The coefficients of polynomials depend upon the extremality parameter \mu >0.In the extremal case \mu = 0 such polynomials were considered previously by H.L\"u, J. Maharana, S. Mukherji and C.N. Pope. Explicit formulas forA_2-solution are obtained. Two examples of A_2-dyon solutions, i.e. dyon in D =11 supergravity with M2 and M5 branes intersecting at a point and Kaluza-Kleindyon, are considered.Comment: 24 pages, Latex, typos are eliminated, a correct relation on  parameters of special block-orthogonal solution is added in third line after  eq. (4.10

Toda p -brane black holes and polynomials related to Lie algebras