Value monoids of zero-dimensional valuations of rank 1

Value Monoids of Zero-dimensional Valuations of Rank One

Classically, Gröbner bases are computed by first prescribing a fixed monomial order. Moss Sweedler suggested an alternative in the mid 1980s 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. We then perform such computations for i...

Realization of a Certain Class of Semi-Groups as Value Semi-Groups of Valuations

Realization of a certain class of semi-groups as value semi-groups of valuations

Given a well-ordered semi-group Γ with a minimal system of generators of ordinal type at most ωn 1 and of rational rank r, which satisfies a positivity and increasing condition, we construct a zero-dimensional valuation centered on the ring of polynomials with r variables such that the semigroup of the values of the polynomial ring is equal to Γ. The construction uses a generalization of Favre ...

On Regularity of Acts

In this article we give a characterization of monoids for which torsion freeness, ((principal) weak, strong) flatness, equalizer flatness or Condition (E) of finitely generated and (mono) cyclic acts and Condition (P) of finitely generated and cyclic acts implies regularity. A characterization of monoids for which all (finitely generated, (mono) cyclic acts are regular will be given too. We als...

Zero Loci of Skew-growth Functions for Dual Artin Monoids

We show that the skew-growth function of a dual Artin monoid of finite type P has exactly rank(P ) =: l simple real zeros on the interval (0, 1]. The proofs for types Al and Bl are based on an unexpected fact that the skewgrowth functions, up to a trivial factor, are expressed by Jacobi polynomials due to a Rodrigues type formula in the theory of orthogonal polynomials. The skew-growth function...