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- W2558177114 abstract "Grobner bases are one of the most powerful tools in computer algebra and commutative algebra, with applications in algebraic geometry and singularity theory. From the theoretical point of view, these bases can be computed over any field using Buchberger's algorithm. In practice, however, the computational efficiency depends on the arithmetic of the coefficient field. In this thesis, we consider Grobner bases computations over two types of coefficient fields. First, consider a simple extension (K=mathbb{Q}(alpha)) of (mathbb{Q}), where (alpha) is an algebraic number, and let (fin mathbb{Q}[t]) be the minimal polynomial of (alpha). Second, let (K') be the algebraic function field over (mathbb{Q}) with transcendental parameters (t_1,ldots,t_m), that is, (K' = mathbb{Q}(t_1,ldots,t_m)). In particular, we present efficient algorithms for computing Grobner bases over (K) and (K'). Moreover, we present an efficient method for computing syzygy modules over (K). To compute Grobner bases over (K), starting from the ideas of Noro [35], 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 [2,4,27], 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 (mathbb{F}_p). This allows us to apply modular methods once again, on a second level, with respect to themodular factors of (f). The algorithm thus resembles a divide and conquer strategy andis in particular easily parallelizable. Moreover, using a similar approach, we present an algorithm for computing syzygy modules over (K). On the other hand, to compute Grobner bases over (K'), our new algorithm first specializes the parameters (t_1,ldots,t_m) to reduce the problem from (K'[x_1,ldots,x_n]) to (mathbb{Q}[x_1,ldots,x_n]). The algorithm then computes a set of Grobner bases of specialized ideals. From this set of Grobner bases with coefficients in (mathbb{Q}), it obtains a Grobner basis of the input ideal using sparse multivariate rational interpolation.At current state, these algorithms are probabilistic in the sense that, as for other modular Grobner 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 algorithms, which have been implemented in SINGULAR [17], are considerably faster than other known methods." @default.
- W2558177114 created "2016-12-08" @default.
- W2558177114 creator A5020785993 @default.
- W2558177114 date "2016-01-01" @default.
- W2558177114 modified "2023-09-23" @default.
- W2558177114 title "Gröbner Bases over Extention Fields of (mathbb{Q})" @default.
- W2558177114 hasPublicationYear "2016" @default.
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