Gerhard Pfister

SINGULAR: a computer algebra system for polynomial computations

SINGULAR is a specialized computer algebra system for polynomial computations with emphasize on the needs of commutative algebra, algebraic geometry, and singularity theory. SINGULAR’s main computational objects are polynomials, ideals and modules over a large variety of rings, including important non-commutative rings. SINGULAR features one of the fastest and most general implementations of various algorithms for computing standard resp. Gröbner bases. Furthermore, it provides multivariate polynomial factorization, resultant, characteristic set and gcd computations, syzygy and free-resolution computations, numerical root–finding, visualization, and many more related functionalities.

Primary Decomposition: Algorithms and Comparisons

The Hilbert series and degree bounds play significant roles in computational invariant theory. In the modular case, neither of these tools is avrulable in general. In this article three results are obtruned, which provide partial remedies for these shortcomings. First, it is shown that the so-called extended Hilbert series, which can always be calculated by a MoHen type formula, yields strong constraints on the degrees of primary invariants. Then it is shown that for a trivial source module the (ordinary) Hilbert series coincides with that of a lift to characteristic 0 and can hence be calculated by MoHen’s formula. The last result is a generalization of Goobel’s degree bound to the case of monomial representations.

Maximal Cohen-Macaulay Modules over the Cone of an Elliptic Curve

Let R=k[Y1,Y2,Y3]/(f), f=Y13+Y23+Y33, where k is an algebraically closed field with chark≠3. Using Atiyah bundle classification over elliptic curves we describe the matrix factorizations of the graded, indecomposable reflexive R-modules, equivalently we describe explicitly the indecomposable bundles over the projective curve V(f)⊂Pk2. Using the fact that over the completion R of R every reflexive module is gradable, we obtain a description of the maximal Cohen–Macaulay modules over R=k〚Y1,Y2,Y3〛/(f).

Local moduli and singularities

The prorepresenting substratum of the formal moduli.- Automorphisms of the formal moduli.- The kodaira-spencer map and its kernel.- Applications to isolated hypersurface singularities.- Plane curve singularities with k*-action.- The generic component of the local moduli suite.- The moduli suite of x 1 5 +x 2 11 .

Identities for finite solvable groups and equations in finite simple groups

We characterise the class of finite solvable groups by two-variable identities in a way similar to the characterisation of finite nilpotent groups by Engel identities. Let u1 = x −2 y −1 x, and un+1 =[ xunx −1 ,y uny −1 ]. The main result states that a finite group G is solvable if and only if for some n the identity un(x, y) ≡ 1h olds inG. We also develop a new method to study equations in the Suzuki groups. We believe that, in addition to the main result, the method of proof is of independent interest: it involves surprisingly diverse and deep methods from algebraic and arithmetic geometry, topology, group theory, and computer algebra (Singular and MAGMA).

Parallelization of Modular Algorithms

In this paper we investigate the parallelization of two modular algorithms. In fact, we consider the modular computation of Grobner bases (resp. standard bases) and the modular computation of the associated primes of a zero-dimensional ideal and describe their parallel implementation in Singular. Our modular algorithms for solving problems over Q mainly consist of three parts: solving the problem modulo p for several primes p, lifting the result to Q by applying the Chinese remainder algorithm (resp. rational reconstruction), and verification. Arnold proved using the Hilbert function that the verification part in the modular algorithm for computing Grobner bases can be simplified for homogeneous ideals (cf. Arnold, 2003). The idea of the proof could easily be adapted to the local case, i.e. for local orderings and not necessarily homogeneous ideals, using the Hilbert-Samuel function (cf. Pfister, 2007). In this paper we prove the corresponding theorem for non-homogeneous ideals in the case of a global ordering.

The Normalization: a new Algorithm, Implementation and Comparisons

We present a new algorithm for computing the normalization \(\bar{R}\) of a reduced affine ring R, together with some remarks on efficiency based on our experience with an implementation of this algorithm in SINGULAR (cf. [2]).

The use of bad primes in rational reconstruction

A standard method for computing a rational number from its values modulo a collection of primes is to determine its value modulo the product of the primes via Chinese Remaindering, and then use Farey sequences for rational reconstruction. This method is guaranteed to work if we restrict ourselves to ”good” primes. Depending on the particular application, however, there is often no efficient way of finding good primes. This note shows that in most situations, we can simply ignore this problem. With regard to applications, we are particularly interested in the design of modular and, thus, parallel versions of algorithms in commutative algebra and algebraic geometry. Here, typically, the final result consists of one or several a priori unknown ideals which are found via constructions yielding the (reduced) Grobner bases of the ideals.

Deformations of maximal Cohen-Macaulay modules

Let (R,m) be a complete Cohen-Macaulay isolated singularity over a field K which is either perfect or [K : K p] 0. According to Dieterich [Di] and [Yo] Ch. 6 (see also [Pol], [PR] or, more generally, [CHP]) there exists a system of parameters x = (xl . . . . . xr) of R such that the base change functor R/(x)®Rdefines an injection v preserving the indecomposability from the set of isomorphism classes of maximal Cohen-Macaulay R-modules to the set of isomorphism classes of R/(x)-modules (in this case the ideal generated by x is called (after [Di]) a reduction ideal. Trying to describe the image of v we noticed in [Po2] that a finitely generated R/(x)module is in Imv if and only if it has the form Xl.. .x~P for a finitely generated R2 := R/(x 2 . . . . . x~ )-module satisfying

Normal forms and moduli spaces of curve singularities with semigroup (2p,2q,2pq+d)

This work has been possible thanks to a Scientific Agreement between the Universidad Complutense and the Humboldt Universitaet. This cooperation agreement supported our stay in the Bereich Algebra and Departamento de Algebra respectively.