Value monoids of zero-dimensional valuations of rank 1

Abstract

Classically, Gröbner bases are computed by first prescribing a set monomial order. Moss Sweedler suggested an alternative and developed a framework to perform such computations by using valuation rings in place of monomial orders. We build on these ideas by providing a class of valuations on k(x, y) that are suitable for this framework.

On the algebraicity of generalized power series

“…• The paper [29] references only the projection from generalized power series in mixed characteristic to positive characteristic, as described in [27,Theorem 7], and is thus unaffected. • The paper [33] makes two references to [26]. One is in the proof of [33,Proposition 5.4], which depends only on the prior result [26, Theorem 1] and is thus unaffected.…”

“…• The paper [33] makes two references to [26]. One is in the proof of [33,Proposition 5.4], which depends only on the prior result [26, Theorem 1] and is thus unaffected. The other is more serious: it is [33, Proposition 5.2], an explicit computation on twist-recurrent series which is then combined with [26,Theorem 8] to deduce [33,Proposition 5.3].…”

“…One is in the proof of [33,Proposition 5.4], which depends only on the prior result [26, Theorem 1] and is thus unaffected. The other is more serious: it is [33, Proposition 5.2], an explicit computation on twist-recurrent series which is then combined with [26,Theorem 8] to deduce [33,Proposition 5.3]. Fortunately, the latter result holds for a simpler reason: for any homomorphism η : Q → K × , the formula i∈Q x i t i → i∈Q η(i)x i t i defines an automorphism of K((t Q )) acting on K((t)), which then also acts on the algebraic closure of K((t)) in K((t Q )) (because any ring homomorphism preserves integral dependence).…”

“…Fortunately, the latter result holds for a simpler reason: for any homomorphism η : Q → K × , the formula i∈Q x i t i → i∈Q η(i)x i t i defines an automorphism of K((t Q )) acting on K((t)), which then also acts on the algebraic closure of K((t)) in K((t Q )) (because any ring homomorphism preserves integral dependence). Consequently, [33] is ultimately unaffected.…”

Reversely well-ordered valuations on rational function fields in two variables

“…However, in [Ke2], Kedlaya constructed a counterexample to a theorem appearing in [Ke1] and proceeds to produce a corrected version. Fortunately, Kedlaya also demonstrates in [Ke2] that the results in [Mo2] remain unaffected by this change. Throughout the remainder of this paper, we turn our attention to the case when the value group is Z ⊕ Z.…”