PRITCHARD, F. L.: On the multiplicity of zeros of polynomials over arbitrary finite dimensional K- algebras, Manuscripta math. 39 (1985), 267-292. 1, 3F. L. PRITCHARD: On the multiplicity of zeros of polynomials over arbitrary finite dimensional K-algebras, Manuscripta math. 49 (1985), ...
Notice first that for any P∈F∩G, the intersection multiplicity of F and G at P must be 1 by Bézout’s theorem. This implies that P is a smooth point of both curves, and that the tangent lines tP(F), tP(G) are different. Second, the polynomials F,G,H,H′ are defined up ...