Some Facts About Canonical Subalgebra Bases

Bravo, Ana

doi:10.1017/cbo9780511756382.009

Cited by 4 publications

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“…Their theory is in many ways similar to the theory of Gröbner bases, with the important difference that not every finitely-generated subalgebra admits a finite SAGBI basis. We refer the reader to Chapter 11 of [Stu96] and to the short survey by Bravo [Bra04] for more information about these concepts.…”

Section: Realizing the Ehrhart Ring As An Initial Subalgebramentioning

confidence: 99%

The Lecture Hall cone as a toric deformation

Katthän

2019

Algebraic and Geometric Combinatorics on Lattice Polytopes

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The Lecture Hall cone is a simplicial cone whose lattice points naturally correspond to Lecture Hall partitions. The celebrated Lecture Hall Theorem of Bousquet-Mélou and Eriksson states that a particular specialization of its multivariate Ehrhart series factors in a very nice and unexpected way. Over the years, several proofs of this result have been found, but it is still not considered to be well-understood from a geometric perspective.In this note we propose two conjectures which aim at clarifying this result. Our main conjecture is that the Ehrhart ring of the Lecture Hall cone is actually an initial subalgebra A n of a certain subalgebra of a polynomial ring, which is itself isomorphic to a polynomial ring. As passing to initial subalgebras does not affect the Hilbert function, this explains the observed factorization. We give a recursive definition of certain Laurent polynomials, which generate the algebra A n . Our second conjecture is that these Laurent polynomials are in fact polynomials.We computationally verified that both conjectures hold for Lecture Hall partitions of length at most 12.Date: September 6, 2018. 1Our indexing of the entries of λ is reversed with respect to the usual convention.

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Section: Realizing the Ehrhart Ring As An Initial Subalgebramentioning

confidence: 99%

The Lecture Hall cone as a toric deformation

Katthän

2019

Algebraic and Geometric Combinatorics on Lattice Polytopes

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“…The ring R is Cohen-Macaulay and its type is less than or equal to #T(o(R)) = 3, where for a numerical semigroup Γ, [ 7,9,19,29, 31 ], [ 7,33 ], [ 4,6,13,15 ], [ 6,8,10,21,23, 25 ], [ 5,18,26,39,47 ], [ 5,18,26,39, 47 ], [ 5,18,26,39,47 ], [ 6,9,19,20 ], [ 7,9,19,29 ], [ 8,9,19, 30 ], [ 7,9,17,19, 29 ] ] Remark 2.18. We do not know a priori ifR = K[[x]] and we do not have a general procedure to check it.…”

Section: Semigroup Of a Formal Space Curvementioning

confidence: 99%

“…For several variables, these basis are known in the literature as SAGBI basis ( [17,4]). Since there are already algorithms in the literature to calculate a basis of A, we will not include the procedure here.…”

Section: Semigroup Of a Polynomial Curvementioning

confidence: 99%

“…The computation of the approximate roots and of Γ(F ) can be performed with the algorithm presented in [2]. gap> f:=y^6-2*x^2*y^3-4*x*y^3-y^3+x^4;; gap> SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity(f); <Numerical semigroup with 3 generators> gap> MinimalGeneratingSystem(last); [ 4,6,9 ] gap> SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity(f); [ [ 6,4,9 ], [ y, y^3-x^2-2*x-1/2 ] ] Example 7.10. Let f (x) = x 6 + x, g(x) = x 4 .…”

Section: Deformation To a Toric Idealmentioning

confidence: 99%

“…The converse is not true in general. Abhyankar asked whether every semigroup generated by a δ-sequence (hence the semigroup of a curve with one place at infinity) is the semigroup of a polynomial curve (for example, the δ-sequence (10,4,5) generates the semigroup 4, 5 which is the semigroup of the polynomial curve A = K[x 4 , x 5 ]). It has been proved recently that the answer is no ( [9]).…”

Section: Basis Of K[f (X) G(x)]mentioning

confidence: 99%

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Bases of subalgebras of K〚x〛 and K[x]

Assi

García-Sánchez

Micale

2017

Journal of Symbolic Computation

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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.

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Monomial Ideals, Binomial Ideals, Polynomial Ideals

Teissier¹

2004

Trends in Commutative Algebra

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Abstract. These lectures provide a glimpse of the applications of toric geometry to singularity theory. They illustrate some ideas and results of commutative algebra by showing the form which they take for very simple ideals of polynomial rings: monomial or binomial ideals, which can be understood combinatorially. Some combinatorial facts are the expression for monomial or binomial ideals of general results of commutative algebra or algebraic geometry such as resolution of singularities or the Briançon-Skoda theorem. In the opposite direction, there are methods that allow one to prove results about fairly general ideals by continuously specializing them to monomial or binomial ideals.

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Some Facts About Canonical Subalgebra Bases

Abstract: Abstract. This is a brief exposition of canonical subalgebra bases, their uses and their computation.

Cited by 4 publications

References 13 publications

The Lecture Hall cone as a toric deformation

The Lecture Hall cone as a toric deformation

Bases of subalgebras of K〚x〛 and K[x]

Monomial Ideals, Binomial Ideals, Polynomial Ideals

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