Factorizations of birational extensions of local rings (Q2382933)

This paper is a proof in a special case of: Abhyankar's local factorization conjecture. Suppose \(R \rightarrow S\) is a birational extension of regular local rings of dimension \(n>2\) and \(V\) is a valuation ring of \(Q(S)\) such that \(V\) dominates \(R\). The conjecture is that there exists a regular local ring \(T\subset V\) such that \(V\) dominates \(T\) and \(T\) dominates both \(R\) and \(S\) and where \(R \rightarrow T\) and \(S \rightarrow T\) are products of monoidal transforms. This conjecture has been already proven in dimension~3 by \textit{S. D. Cutkosky} [Adv. Math. 132, No. 2, 167--315 (1997; Zbl 0934.14006)]. In this new paper, \(R\subset S\) are regular local rings essentially of finite type over a field \(k\) of dimension \(n>2\) of characteristic~0, Abhyankar's conjecture is proven in the case where the valuation \(v\) has rank~1 and maximal rational rank: the valuation group \(VQ(S)\subset \mathbb R\) and has rational rank \(n\). By a Cutkosky's theorem, we may suppose that \(R\subset S\) is monomial, i.e. there exist a regular system of parameters \((x_1,\dots,x_n)\) (resp. \((y_1,\dots,y_n)\)) of \(R\) (resp. \(S\)) such that \[ x_i=\prod_{1\leq j \leq n} y_j^{a_{ij}}. \] Let us define \(\overrightarrow{v}:=(v(x_1),\dots,v(x_n))^t\), \(\overrightarrow{w}:=(v(y_1),\dots,v(y_n))^t\), \(A:=(a_{ij})\): \(A\) is a \(n\times n\) matrix with coefficients in \(\mathbb N\), \(\overrightarrow{v}\) and \(\overrightarrow{w}\) are real vectors with strictly positive rationally independant coefficients. Then, it appears that, if you blow up \(S\) along \((y_1,y_2)\), you make ``permissible column addition'' on \(A\) and \(\overrightarrow{w}\): you just add the second column to the first if \(v(y_2)>v(y_1)\), and in \(\overrightarrow{w}\), you change \(w_2\)in \(w_2-w_1\). If you you blow up \(R\) along \((x_1,x_2)\) and that \(S(1)\) the localization of this b.u. at the center of \(v\) is dominated by \(S\), then you make to \(A\) and \(\overrightarrow{v}\) ``permissible row substractions''. The authors prove that by a sequence of ``permissible column additions'', row interchanges and ``permissible row substractions'' you can make \(A=\text{Id}\). When you translate in the language of blowing ups, it means that you reach \(T\) from \(S\) and \(R\) by very special monoidal transforms. The authors do not pretend to originality in the result: there is already a proof of \textit{K. Karu} [J. Algebr. Geom. 14, No. 1, 165--175 (2005; Zbl 1077.14017)]. But their proof (inspired by Christensen) is the most elementary I ever read in this topic and is understandable by any undergraduate student in mathematics. The reviewer has the feeling that, with the same techniques, it is possible to solve the following problem. Problem. Given \(R\subset S\) monomial, can you find a CNS condition on \((A,\overrightarrow{v},\overrightarrow{w})\) such that \(R\subset S\) is a product of monoidal transforms?