Abstract: Although Buchberger's 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(a), a simple extension of Q, where a is an algebraic number, and let f be the minimal polynomial of a over Q. In this talk 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 (M. Noro: An efficient implementation for computing Gröbner bases over algebraic number fields. In Mathematical software - ICMS 2006. Second international congress on mathematical software, Castro Urdiales, Spain, September 1-3, 2006. Proceedings, pages 99-109. Springer, 2006), 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 (E. A. Arnold. Modular algorithms for computing Gröbner bases. J. Symb. Comput., 35(4):403-419, 2003., J. Böhm, W. Decker, C. Fieker, and G. Pfister. The use of bad primes in rational reconstruction. Math. Comp. 84 (2015), 3013-3027, N. Idrees, G. Pfister, and S. Steidel. Parallelization of modular algorithms. J. Symb. Comput., 46(6):672-684, 2011.), 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 F_p. 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, outperforms other known methods by far.
Abstract: Computations over the rational numbers often suffer from intermediate coefficient growth. One approach to this problem is to determine the result modulo a number of primes and then lift 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. We develop a new technique for rational reconstruction which will nevertheless return the correct result, provided the number of good primes in the set is large enough. We discuss applications of this technique in computational algebraic geometry. This is joint work with Claus Fieker, Wolfram Decker, Gerhard Pfister, and Santiago Laplagne.
Abstract: Resolution of singularities was proved by Hironaka in the case of characteristic zero in the 1960s, but in positive characteristic it still is one of the longstanding open problems in Algebraic Geometry. The study of this problem has led to interesting questions some related to number theory, some to combinatorics and yet others to algorithmic considerations. In this talk I shall describe the general structure of Hironaka-style desingularization algorithms, explain the main difficulties in positive characteristic and give a brief overview on results in low dimensions.
Abstract: Singular is a comprehensive and steadily growing computer algebra system, with particular emphasis on applications in algebraic geometry, commutative algebra, and singularity theory. Singular provides highly efficient core algorithms, a multitude of advanced algorithms in the above fields, an intuitive, C-like programming language, i easy ways to make it user-extendable through libraries, and a comprehensive online manual and help function. Singular's core algorithms handle Gr&oml;bner resp. standard bases and free resolutions, polynomial factorization, resultants, characteristic sets, and numerical root finding. Its advanced algorithms, contained in its libraries, address topics such as absolute factorization, algebraic D-modules, classification of singularities, deformation theory, Gauss-Manin systems, Hamburger-Noether (Puiseux) development, invariant theory, homological algebra, normalization, primary decomposition, resolution of singularities, and sheaf cohomology. These properties allow Singular to be used as a valuable tool in all areas which need to handle large polynomial systems, but often it needs to be combined with other tools. In this talk I will present Singular's possibilities as a C++ library which allows it to be included in other systems and discuss the current use cases of libSingular in GAP, polymake and SAGE. On the other side: Singular can also be easily extend by other tool provided as C or C++ library: gfan will be discussed as an example.