Generic incarnations of quadratic transforms (Q2855889)

The author prove a generalization of the Bézout theorem about the multiplicity of intersection of two curves. More precisely the result given here is valid for two nonzero elements in the maximal ideal of a two dimensional regular local domain \(R\). The proof consists before to reduce the problem to one dimensional local rings and after to zero dimensional local rings. The essential tool of the proof is what the author call the generic incarnation of the set of all the regular domains \(S\) which birationally dominate \(R\) : this is the set of all the \(S [t]\) localized in a the multiplicative set of all those member of \(S [t]\) whose coefficients generate \(S\); with \(S\) a regular domain which birationally dominate \(R\) (see \textit{S. S. Abhyankar} [Proc. Am. Math. Soc. 140, No. 12, 4111--4126 (2012; Zbl 1307.14001)] and [\textit{S. S. Abhyankar}, ``Dicritical Divisors and Dedekinds Gauss Lemma'', Revista Mathematica Complutense (2012)]).NEWLINENEWLINE In the last sections the author give a proof of the theorem (4.6) of [\textit{S. S. Abhyankar} and \textit{W. J. Heinzer}, Proc. Indian Acad. Sci., Math. Sci. 122, No. 4, 525--546 (2012; Zbl 1275.13015)] about the commutativity of the contact number introduce in [Proc. Am. Math. Soc. 139, No. 9, 3067--3082 (2011; Zbl 1227.14004)] but this time without assuming that the ideals have 2-generetad reductions, i.e.: there exists two nonzero elements of the maximal ideal of \(R\) such that the integral closure in \(R\) of the ideal generated by this two elements is the starting ideal.