Wolfram Decker

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.

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]).

Syzygies of Abelian and Bielliptic Surfaces in ℙ4

So far only six families of smooth irregular surfaces are known to exist in P^4 (up to pullbacks by suitable finite covers of P^4). These are the elliptic quintic scrolls, the minimal abelian and bielliptic surfaces (of degree 10), two different families of non-minimal abelian surfaces of degree 15, and one family of non-minimal bielliptic surfaces of degree 15. The main purpose of the paper is to describe the structure of the Hartshorne-Rao modules and the syzygies for each of these smooth irregular surfaces in P^4, providing at the same time a unified construction method (via syzygies) for these families of surfaces.

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.

Parallel algorithms for normalization

Given a reduced affine algebra A over a perfect field K, we present parallel algorithms to compute the normalization A@? of A. Our starting point is the algorithm of Greuel et al. (2010), which is an improvement of de Jong@?s algorithm (de Jong, 1998; Decker et al., 1999). First, we propose to stratify the singular locus Sing(A) in a way which is compatible with normalization, apply a local version of the normalization algorithm at each stratum, and find A@? by putting the local results together. Second, in the case where K=Q is the field of rationals, we propose modular versions of the global and local-to-global algorithms. We have implemented our algorithms in the computer algebra system Singular and compare their performance with that of the algorithm of Greuel et al. (2010). In the case where K=Q, we also discuss the use of modular computations of Grobner bases, radicals, and primary decompositions. We point out that in most examples, the new algorithms outperform the algorithm of Greuel et al. (2010) by far, even if we do not run them in parallel.

Comparison of probabilistic algorithms for analyzing the components of an affine algebraic variety

Systems of polynomial equations arise throughout mathematics, engineering, and the sciences. It is therefore a fundamental problem both in mathematics and in application areas to find the solution sets of polynomial systems. The focus of this paper is to compare two fundamentally different approaches to computing and representing the solutions of polynomial systems: numerical homotopy continuation and symbolic computation. Several illustrative examples are considered, using the software packages Bertini and Singular.

Generating a Noetherian Normalization of the Invariant Ring of a Finite Group

We improve Kempers algorithm for the computation of a noetherian normalization of the invariant ring of a finite group. The new algorithm, which still works over arbitrary coefficient fields, no longer relies on algorithms for primary decomposition.

Local to global algorithms for the Gorenstein adjoint ideal of a curve

We present new algorithms for computing adjoint ideals of curves and thus, in the planar case, adjoint curves. With regard to terminology, we follow Gorenstein who states the adjoint condition in terms of conductors. Our main algorithm yields the Gorenstein adjoint ideal \(\mathfrak {G}\) of a given curve as the intersection of what we call local Gorenstein adjoint ideals. Since the respective local computations do not depend on each other, our approach is inherently parallel. Over the rationals, further parallelization is achieved by a modular version of the algorithm which first computes a number of the characteristic p counterparts of \(\mathfrak {G}\) and then lifts these to characteristic zero. As a key ingredient, we establish an efficient criterion to verify the correctness of the lift. Well-known applications are the computation of Riemann-Roch spaces, the construction of points in moduli spaces, and the parametrization of rational curves. We have implemented different variants of our algorithms together with Mnuk’s approach in the computer algebra system Singular and give timings to compare the performance.

Bad Primes in Computational Algebraic Geometry

Computations over the rational numbers often suffer from intermediate coefficient swell. One solution to this problem is to apply the given algorithm modulo a number of primes and then lift the modular results to the rationals. This method is guaranteed to work if we use a sufficiently large set of good primes. In many applications, however, there is no efficient way of excluding bad primes. In this note, we describe a technique for rational reconstruction which will nevertheless return the correct result, provided the number of good primes in the selected set of primes is large enough. We give a number of illustrating examples which are implemented using the computer algebra system Singular and the programming language Julia. We discuss applications of our technique in computational algebraic geometry.

Gröbner bases over algebraic number fields

Although Buchbergers algorithm, in theory, allows us to compute Gröbner bases over any field, in practice, however, the computational efficiency depends on the arithmetic of the ground field. Consider a field K = Q(α), a simple extension of Q, where α is an algebraic number, and let f ∈ Q[t] be the minimal polynomial of α. In this paper we present a new efficient method to compute Gröbner bases in polynomial rings over the algebraic number field K. Starting from the ideas of Noro [11], we proceed by joining f to the ideal to be considered, adding t as an extra variable. But instead of avoiding superfluous S-pair reductions by inverting algebraic numbers, we achieve the same goal by applying modular methods as in [2, 3, 10], that is, by inferring information in characteristic zero from information in characteristic p > 0. For suitable primes p, the minimal polynomial f is reducible over Fp. This allows us to apply modular methods once again, on a second level, with respect to the factors of f. The algorithm thus resembles a divide and conquer strategy and is in particular easily parallelizable. At current state, the algorithm is probabilistic in the sense that, as for other modular Gröbner basis computations, an effective final verification test is only known for homogeneous ideals or for local monomial orderings. The presented timings show that for most examples, our algorithm, which has been implemented in Singular [7], outperforms other known methods by far.