Bases of subalgebras of K〚x〛 and K[x]

Abstract

Let f1, . . . , fs be formal power series (respectively polynomials) in the variable x. We study the semigroup of orders of the formal series in the algebra K f1, . . . , fs ⊆ K x (respectively the semigroup of degrees of polynomials in K[f1, . . . , fs] ⊆ K[x]). We give procedures to compute these semigroups and several applications. We prove in particular that the space curve parametrized by f1, . . . , fs has a flat deformation into a monomial curve.Proof. This follows easily from the definition of order. Proposition 2.2. [13, Lemma 3, p.486] Let R 1 and R 2 be rings of our type such that R 1 ⊆ R 2 and o(R 1 ) = o(R 2 ). Then R 1 = R 2 . Proposition 2.3. [13, Proposition 1, p.488] Let R be a ring of our type. Then λ R (R/R) = |N \ o(R)|.The following two results appear in [15], and since this paper is very hard to find, we include the proofs for sake of completeness. Proposition 2.4. [15] Let R be a ring of our type. Then o(R) is a numerical semigroup.Proof. Since R is a ring and by (iii) of Proposition 2.

numericalsgps, a GAP package for numerical semigroups

“…If (a, b) is one of this minimal generators, then ϕ(a) = ϕ(b) ∈ S is called a primitive element of S. These elements play an important role in factorization properties of S, and consequently we provide a function to compute them. gap> s:=NumericalSemigroup (5,7,9); <Numerical semigroup with 3 generators> gap> MinimalPresentationOfNumericalSemigroup(s);…”

“…Also there is a procedure to compute all numerical semigroups with given Frobenius number (this is done using the concept of fundamental gap as explained in [43]) and another function to compute the set of all numerical semigroups with given genus g (by constructing the tree of all numerical semigroups up to the level g). gap> s:=NumericalSemigroup (5,7,9); <Numerical semigroup with 3 generators> gap> Length(OverSemigroupsNumericalSemigroup(s)); 15 gap> Length(NumericalSemigroupsWithFrobeniusNumber (21)); 1828…”

“…We give a function to test if a numerical semigroup is almost-symmetric and include the procedure presented in [41] to compute the set of almost symmetric numerical semigroups with fixed Frobenius number. gap> s:=NumericalSemigroup (5,7,9); <Numerical semigroup with 3 generators> gap> DecomposeIntoIrreducibles(s); [ <Numerical semigroup>, <Numerical semigroup> ] gap> List(last,MinimalGeneratingSystem); [ [ 5,7,8,9 ], [ 5,7,9,11 ] ] gap> Length(TelescopicNumericalSemigroupsWithFrobeniusNumber(101)); 86 gap> Length(AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber (31) We provide functions for computing the small elements of an ideal (the definition is analogous to that in numerical semigroups), Apéry sets (and tables; see [20]), the ambient numerical semigroup, membership, and also some basic operations as addition, union, subtraction (I − J = {z ∈ Z | z + J ⊆ I}), set difference, multiplication by an integer, translation by an integer, intersection, blow-up ( n∈N nI − nI) and * -closure with respect to a family of ideals ( [44]).…”

“…Then the set of values (respectively degrees) of the series (respectively polynomials) in the algebra K F (respectively K[F]) is a submonoid of N. Under certain conditions it is a numerical semigroup, and we provide functions to compute it. Also to determine a basis of the algebra K F (or K[F]) such that the values (or degrees) minimally generate the semigroup of values of this algebra (see [5]). gap> t:=Indeterminate(Rationals,"t");; gap> l:=[tˆ4,tˆ6+tˆ7,tˆ13]; [ tˆ4, tˆ7+tˆ6, tˆ13 ] gap> SemigroupOfValuesOfCurve_Local(l); <Numerical semigroup with 4 generators> gap> MinimalGeneratingSystem(last); [ 4,6,13,15 ] gap> SemigroupOfValuesOfCurve_Local(l,"basis"); [ tˆ4, tˆ7+tˆ6, tˆ13, tˆ15 ] 2.11.…”

Canonical bases of modules over one dimensional k-algebras

“…, F r (t) ∈ K[t]). This algorithm uses the one given in [5] in order to compute the semigroup consisting of degrees in t of elements of A. Then we consider the case where A = K[X(t), Y (t)] is the ring of coordinates of the algebraic plane curve parametrized by X(t), Y (t), and K is an algebraically closed field of characteristic zero.…”