Degree bounds for Gröbner bases in algebras of solvable type

We establish doubly-exponential degree bounds for Gröbner bases in certain algebras of solvable type over a field (as introduced by Kandri-Rody and Weispfenning). The class of algebras considered here includes commutative polynomial rings, Weyl algebras, and universal enveloping algebras of finite-dimensional Lie algebras. For the computation of these bounds, we adapt a method due to Dubé based on a generalization of Stanley decompositions. Our bounds yield doubly-exponential degree bounds for ideal membership and syzygies, generalizing the classical results of Hermann and Seidenberg (in the commutative case) and Grigoriev (in the case of Weyl algebras).     

Gröbner bases of ideals cogenerated by Pfaffians

We characterise the class of one-cogenerated Pfaffian ideals whose natural generators form a Gröbner basis with respect to any anti-diagonal term order. We describe their initial ideals as well as the associated simplicial complexes, which turn out to be shellable and thus Cohen-Macaulay. We also provide a formula for computing their multiplicity.     

New developments in the theory of Gröbner bases and applications to formal verification

We present foundational work on standard bases over rings and on Boolean Gröbner bases in the framework of Boolean functions. The research was motivated by our collaboration with electrical engineers and computer scientists on problems arising from formal verification of digital circuits. In fact, algebraic modelling of formal verification problems is developed on the word-level as well as on the bit-level. The word-level model leads to Gröbner basis in the polynomial ring over Z/2ⁿ while the bit-level model leads to Boolean Gröbner bases. In addition to the theoretical foundations of both approaches, the algorithms have been implemented. Using these implementations we show that special data structures and the exploitation of symmetries make Gröbner bases competitive to state-of-the-art tools from formal verification but having the advantage of being systematic and more flexible.     

On the Gröbner complexity of matrices

In this paper we show that if for an integer matrix A the universal Gröbner basis of the associated toric ideal I_A coincides with the Graver basis of A, then the Gröbner complexity u(A) and the Graver complexity g(A) of its higher Lawrence liftings agree, too. In fact, if the universal Gröbner basis of I_A coincides with the Graver basis of A, then also the more general complexities u(A,B) and g(A,B) agree for arbitrary B. We conclude that for the matrices A_3×3 and A_3×4, defining the 3×3 and 3×4 transportation problems, we have u(A_3×3)=g(A_3×3)=9 and u(A_3×4)=g(A_3×4)≥27. Moreover, we prove that u(A_a,b)=g(A_a,b)=2(a+b)/gcd(a,b) for positive integers a,b and .     

Gröbner-Shirshov bases for associative algebras with multiple operators and free Rota-Baxter algebras

In this paper, we establish the Composition-Diamond lemma for associative algebras with multiple linear operators. As applications, we obtain Gröbner-Shirshov bases of free Rota-Baxter algebra, free λ-differential algebra and free λ-differential Rota-Baxter algebra, respectively. In particular, linear bases of these three free algebras are respectively obtained, which are essentially the same or similar to the recent results obtained by K. Ebrahimi-Fard-L. Guo, and L. Guo-W. Keigher by using other methods.     

Applying Gröbner basis techniques to group theory

We apply the machinery of Gröbner bases to finitely presented groups. This allows for computational methods to be developed which prove that a given finitely presented group is not n-linear over a field k assuming some mild conditions. We also present an algorithm which determines whether or not a finitely presented group G is trivial given that an oracle has told us that G is n-linear over an algebraically closed field k.     

Improving the bounds of the multiplicity conjecture: The codimension 3 level case

The Multiplicity Conjecture (MC) of Huneke and Srinivasan provides upper and lower bounds for the multiplicity of a Cohen-Macaulay algebra A in terms of the shifts appearing in the modules of the minimal free resolution (MFR) of A. All the examples studied so far have lead to conjecture (see [J. Herzog, X. Zheng, Notes on the multiplicity conjecture. Collect. Math. 57 (2006) 211-226] and [J. Migliore, U. Nagel, T. Römer, Extensions of the multiplicity conjecture, Trans. Amer. Math. Soc. (preprint: math.AC/0505229) (in press)]) that, moreover, the bounds of the MC are sharp if and only if A has a pure MFR. Therefore, it seems a reasonable-and useful-idea to seek better, if possibly ad hoc, bounds for particular classes of Cohen-Macaulay algebras.In this work we will only consider the codimension 3 case. In the first part we will stick to the bounds of the MC, and show that they hold for those algebras whose h-vector is that of a compressed algebra.In the second part, we will (mainly) focus on the level case: we will construct new conjectural upper and lower bounds for the multiplicity of a codimension 3 level algebra A, which can be expressed exclusively in terms of the h-vector of A, and which are better than (or equal to) those provided by the MC. Also, our bounds can be sharp even when the MFR of A is not pure.Even though proving our bounds still appears too difficult a task in general, we are already able to show them for some interesting classes of codimension 3 level algebras A: namely, when A is compressed, or when its h-vector h(A) ends with (…,3,2). Also, we will prove our lower bound when h(A) begins with (1,3,h₂,…), where h₂≤4, and our upper bound when h(A) ends with (…,h_c−1,h_c), where h_c−1≤h_c+1.     

The symmetric group given by a Gröbner basis

We present a Gröbner basis associated with the symmetric group of degree n, which is determined by a strong generating set of the symmetric group and is defined by means of a term ordering with the elimination property.     

Degrevlex Gröbner bases of generic complete intersections

In this paper, we study the Hilbert–Samuel function of a generic standard graded K-algebra

K[X₁,…,X_n]/(g₁,…,g_m)

when refined by an (ℓ)-adic filtration, ℓ being a linear form. From this we obtain a structure theorem which describes the stairs of a generic complete intersection for the degree-reverse-lexicographic order. We show what this means for generic standard (or Gröbner) bases for this order; in particular, we consider an “orderly filling up” conjecture, and we propose a strategy for the standard basis algorithm which could be useful in generic-like cases.     

Normal toric ideals of low codimension

Every normal toric ideal of codimension two is minimally generated by a Gröbner basis with squarefree initial monomials. A polynomial time algorithm is presented for checking whether a toric ideal of fixed codimension is normal.     

Radical cube zero weakly symmetric algebras and support varieties

One of our main results is a classification of all the weakly symmetric radical cube zero finite dimensional algebras over an algebraically closed field having a theory of support via the Hochschild cohomology ring satisfying Dade’s Lemma. In the process we give a characterization of when a finite dimensional Koszul algebra has such a theory of support in terms of the graded centre of the Koszul dual.     

Radical cube zero selfinjective algebras of finite complexity

One of our main results is a classification of all the possible quivers of selfinjective radical cube zero finite-dimensional algebras over an algebraically closed field having finite complexity. In the paper (Erdmann and Solberg, 2011) [5] we classified all weakly symmetric algebras with support varieties via Hochschild cohomology satisfying Dade’s Lemma. For a finite-dimensional algebra to have such a theory of support varieties implies that the algebra has finite complexity. Hence this paper is a partial extension of [5].     

Algebraic solutions of plane vector fields

We present an algorithm that can be used to check whether a given derivation of the complex affine plane has an invariant algebraic curve and discuss the performance of its implementation in the computer algebra system Singular.     

Int-decomposable algebras

Let D be a Dedekind domain with fraction field k. Let A be a D-algebra that, as a D-module, is free of finite rank. Let B be the extension of A to a k-algebra. The set of integer-valued polynomials over A   is defined to be Int(A)={f∈B[x]|f(A)⊆A}$\mathrm{Int}(A)=\{f\in B[x]|f(A)\subseteq A\}$. Restricting the coefficients to elements of k  , we obtain the commutative ring _Intk(A)={f∈k[x]|f(A)⊆A}${\mathrm{Int}}_k(A)=\{f\in k[x]|f(A)\subseteq A\}$; this makes Int(A)$\mathrm{Int}(A)$ a left _Intk(A)${\mathrm{Int}}_k(A)$-module. Previous researchers have noted instances when a D-module basis for A   is also an _Intk(A)${\mathrm{Int}}_k(A)$-basis for Int(A)$\mathrm{Int}(A)$. We classify all the D-algebras A   with this property. Along the way, we prove results regarding Int(A)$\mathrm{Int}(A)$, its localizations at primes of D, and finite residue rings of A.     

Computing border bases

This paper presents several algorithms that compute border bases of a zero-dimensional ideal. The first relates to the FGLM algorithm as it uses a linear basis transformation. In particular, it is able to compute border bases that do not contain a reduced Gröbner basis. The second algorithm is based on a generic algorithm by Bernard Mourrain originally designed for computing an ideal basis that need not be a border basis. Our fully detailed algorithm computes a border basis of a zero-dimensional ideal from a given set of generators. To obtain concrete instructions we appeal to a degree-compatible term ordering σ and hence compute a border basis that contains the reduced σ-Gröbner basis. We show an example in which this computation actually has advantages over Buchberger's algorithm. Moreover, we formulate and prove two optimizations of the Border Basis Algorithm which reduce the dimensions of the linear algebra subproblems.     

Weighted locally gentle quivers and Cartan matrices

We introduce and study the class of weighted locally gentle quivers. This naturally extends the class of gentle quivers and gentle algebras, which have been intensively studied in the representation theory of finite-dimensional algebras, to a wider class of potentially infinite-dimensional algebras. Weights on the arrows of these quivers lead to gradings on the corresponding algebras. For natural grading by path lengths, any locally gentle algebra is Koszul. The class of locally gentle algebras consists of the gentle algebras together with their Koszul duals.Our main result is a general combinatorial formula for the determinant of the weighted Cartan matrix of a weighted locally gentle quiver. We show that this weighted Cartan determinant is a rational function which is completely determined by the combinatorics of the quiver-more precisely by the number and the weight of certain oriented cycles.     

The geometry of border bases

In this paper we continue the study of the border basis scheme we started in Kreuzer and Robbiano (2008) [16]. The main topic is the construction of various explicit flat families of border bases. To begin with, we cover the punctual Hilbert scheme by border basis schemes and work out the base changes. This enables us to control flat families obtained by linear changes of coordinates. Next we provide an explicit construction of the principal component of the border basis scheme, and we use it to find flat families of maximal dimension at each radical point. Finally, we connect radical points to each other and to the monomial point via explicit flat families on the principal component.     

Vanishing of self-extensions over symmetric algebras

We study self-extensions of modules over symmetric artin algebras. We show that non-projective modules with eventually vanishing self-extensions must lie in AR components of stable type Z_A∞$\mathbb{Z}{A}_{\infty }$. Moreover, the degree of the highest non-vanishing self-extension of these modules is determined by their quasilength. This has implications for the Auslander–Reiten Conjecture and the Extension Conjecture.