Springer Proceedings in Mathematics & Statistics, 2014
In this article we present a parallel modular algorithm to compute all solutions of a given zero-... more In this article we present a parallel modular algorithm to compute all solutions of a given zero-dimensional polynomial system of equations over the rationals. In fact, we compute a triangular decomposition of the corresponding ideal in the polynomial ring over the rationals using modular methods, and then apply a solver for univariate polynomials.
Given a reduced affine algebra A over a perfect field K, we present parallel algorithms to comput... more 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 , which is an improvement of de Jong's algorithm, see . 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 algorithms. We have implemented our algorithms in the computer algebra system Singular and compare their performance with that of other algorithms. In the case where K = Q, we also discuss the use of modular computations of Gröbner bases, radicals and primary decompositions. We point out that in most examples, the new algorithms outperform the algorithm of ] by far, even if we do not run them in parallel.
In this paper we investigate the parallelization of two modular algorithms. In fact, we consider ... more In this paper we investigate the parallelization of two modular algorithms. In fact, we consider the modular computation of Gröbner 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 to solve problems over Q mainly consist of three parts, solving the problem modulo p for several primes p, lifting the result to Q by applying Chinese remainder resp. rational reconstruction, and a part of verification. Arnold proved using the Hilbert function that the verification part in the modular algorithm to compute Gröbner bases can be simplified for homogeneous ideals (cf.
We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over ... more We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals resp. over finite fields, and the idea of Shimoyama-Yokoyama resp. Eisenbud-Hunecke-Vasconcelos to extract primary ideals from pseudo-primary ideals. A parallelized version of the algorithm is implemented in Singular. Examples and timings are given at the end of the article.
In this article we describe our experiences with a parallel Singular-implementation of the signat... more In this article we describe our experiences with a parallel Singular-implementation of the signature of a surface singularity defined by z N + g(x, y) = 0.
Springer Proceedings in Mathematics & Statistics, 2014
In this article we present a parallel modular algorithm to compute all solutions of a given zero-... more In this article we present a parallel modular algorithm to compute all solutions of a given zero-dimensional polynomial system of equations over the rationals. In fact, we compute a triangular decomposition of the corresponding ideal in the polynomial ring over the rationals using modular methods, and then apply a solver for univariate polynomials.
Given a reduced affine algebra A over a perfect field K, we present parallel algorithms to comput... more 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 , which is an improvement of de Jong's algorithm, see . 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 algorithms. We have implemented our algorithms in the computer algebra system Singular and compare their performance with that of other algorithms. In the case where K = Q, we also discuss the use of modular computations of Gröbner bases, radicals and primary decompositions. We point out that in most examples, the new algorithms outperform the algorithm of ] by far, even if we do not run them in parallel.
In this paper we investigate the parallelization of two modular algorithms. In fact, we consider ... more In this paper we investigate the parallelization of two modular algorithms. In fact, we consider the modular computation of Gröbner 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 to solve problems over Q mainly consist of three parts, solving the problem modulo p for several primes p, lifting the result to Q by applying Chinese remainder resp. rational reconstruction, and a part of verification. Arnold proved using the Hilbert function that the verification part in the modular algorithm to compute Gröbner bases can be simplified for homogeneous ideals (cf.
We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over ... more We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals resp. over finite fields, and the idea of Shimoyama-Yokoyama resp. Eisenbud-Hunecke-Vasconcelos to extract primary ideals from pseudo-primary ideals. A parallelized version of the algorithm is implemented in Singular. Examples and timings are given at the end of the article.
In this article we describe our experiences with a parallel Singular-implementation of the signat... more In this article we describe our experiences with a parallel Singular-implementation of the signature of a surface singularity defined by z N + g(x, y) = 0.
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Papers by Stefan Steidel