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Abstract. In this paper we derive an upper bound for the degree of the strict invariant algebraic curve of a polynomial system in the complex project plane under generic condition. The results are obtained through the algebraic multiplicities of the system at the singular points. A method for computing the algebraic multiplicity using Newton polygon is also presented.

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Consideration of the monodromy group of the hypergeometric equation z(1−z)w″+[γ−(1+α+β)z]w′−αβw=0, in the case of α=1/6, β=5/6, γ=7/6, shows that the global hypergeometric function solution F(1/6;5/6;7/6;z) is nonalgebraic although it has only algebraic singularities. Therefore, the proposition given in [2,4] that a function is algebraic if it has only the algebraic singularities on the extended z-plane is not true. Through introduction of the concept of singular element criterion for deciding when a function is algebraic on the basis of properties of its singularities is given