Residual ideals of MacLane valuations (Q2259148)

The authors build on results of \textit{S. MacLane} [Proc. Natl. Acad. Sci. USA 21, 472--474 (1935; Zbl 0012.09903); Duke Math. J. 2, 492--510 (1936; Zbl 0015.05801); Trans. Am. Math. Soc. 40, 363--395 (1936; Zbl 0015.29202)] and \textit{M. Vaquié} [Trans. Am. Math. Soc. 359, No. 7, 3439--3481 (2007; Zbl 1121.13006); J. Algebra 311, No. 2, 859--876 (2007; Zbl 1121.13007); Ann. Inst. Fourier 58, No. 7, 2503--2541 (2008; Zbl 1170.13003)], while restricting their considerations to discrete valuations \((K,v)\) on a field \(K\). Their approach is through a study of residual ideals of the graded algebra and their inspiration and motivation are computational applications. Among a number of results, the authors determine the structure of the graded algebra of the discrete valuations on the rational functions field \(K(x)\). The second path the authors explore is to analyze the structure of the set \(P\) of prime polynomials with respect to \(v\), namely monic irreducible polynomials in \(\mathcal O_v[x]\), where \(\mathcal O_v\) is the valuation ring of the completion \(K_v\) of \(K\) at \(v\). The constructive methods of the paper lead to the main result which establishes a canonical bijection between the quotient set \(P/\approx\) (the Okutsu equivalent classes of prime polynomials) and a MacLane space \(M\), defined as the set of pairs \((\mu, \mathcal L)\), where \(\mu\) is an inductive valuation on \(K(x)\) and \(\mathcal L\) is a strong maximal ideal of the degree-zero subring \(\Delta(\mu)\) of \(\mathcal Gr(\mu)\); the bijection sends the class of \(F\) to the pair \((\mu_F, R_{\mu F}(F))\). A point in the space \(M\) is described in terms of discrete parameters, which may be considered as a kind of genetic sequence that is common to all prime polynomials in the corresponding Okutsu class.