ALEXANDER POLYNOMIALS OF PLANE ALGEBRAIC CURVES

Abstract

The author studies the fundamental group of the complement of an algebraic curve defined by an equation . Let be the morphism defined by the equation . The main result is that if the generic fiber is irreducible, then the kernel of the homomorphism is a finitely generated group. In particular, if is an irreducible curve, then the commutator subgroup of is finitely generated.The internal and external Alexander polynomials of a curve (denoted by and respectively) are introduced, and it is shown that the Alexander polynomial of the curve divides and and is a reciprocal polynomial whose roots are roots of unity. Furthermore, if is an irreducible curve, the Alexander polynomial of the curve satisfies the condition . From this it follows that among the roots of the Alexander polynomial of an irreducible curve there are no primitive roots of unity of degree , where is a prime number.