F. L. PRITCHARD: On the multiplicity of zeros of polynomials over arbitrary finite dimensional K-algebras, Manuscripta math. 49 (1985), 267-292. 2PRITCHARD, F. L.: On the multiplicity of zeros of polynomials over arbitrary finite dimensional K- algebras, Manuscripta math. 39 (1985), 267-...
. . , yn ∈ R which are polynomials in pm+1- th powers of elements of R, such that the map Spec R −→ Ank given by x1 + y1, . . . , xn + yn is finite. Let yi be any lifts of yi to R◦ which are polynomials in pm+1-powers of elements of R◦. Arguing as in...
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 ...