Computations on Algebraic Curves and Surfaces [SFB F1304-1]

@techreport{RISC5178,
author = {Christoph Fuerst and Guenter Landsmann},
title = {{Three Examples of Gröbner Reduction over Noncommutative Rings}},
language = {english},
abstract = {In this report three classes of noncommutative rings are investigated withemphasis on their properties with respect to reduction relations. TheGröbner basis concepts in these rings, being developed in the literature byseveral authors, are considered and it is shown that the reduction relationscorresponding to these Gröbner bases obey the axioms of a general theoryof Gröbner reduction.},
number = {15-16},
year = {2015},
month = {October},
sponsor = {partially supported by the Austrian Science Fund (FWF): W1214-N15, project DK11},
length = {31},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}

@techreport{RISC5068,
author = {Christoph Fuerst and Guenter Landsmann},
title = {{The Concept of Gröbner Reduction for Dimension in filtered free modules}},
language = {english},
abstract = {We present the concept of Gröbner reduction that is a Gröbner basistechnique on filtered free modules. It allows to compute the dimensionof a filtered free module viewn as a K-vector space. We apply the de-veloped technique to the computation of a generalization of Hilbert-typedimension polynomials in K[X] as well as in finitely generated difference-differential modules. The latter allows us to determine a multivariatedimension polynomial where we partition the set of derivations and theset of automorphism in a difference-differential ring in an arbitrary way.},
number = {14-12},
year = {2014},
month = {October},
length = {13},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}

@techreport{RISC3042,
author = {Meng Zhou and Franz Winkler},
title = {{Computing difference-differential Groebner Bases and difference-differential dimension polynomials}},
language = {english},
abstract = {Dfference-differential Groebner bases and the algorithms were introduced byM.Zhou and F.Winkler (2006). In this paper we will make further investigationsfor the key concept of S-polynomials in the algorithm and we will improve tech-nically the algorithm. Then we apply the algorithm to compute the difference-differential dimension polynomial of a difference-differential module and of asystem of linear partial difference-differential equations. Also, in cyclic modulecase, we present an algorithm to compute the difference-differential dimensionpolynomials in two variables with the Groebner basis.},
number = {07-01},
year = {2007},
month = {January},
keywords = {difference-differential Groebner basis, S-polynomial, difference-differential dimension polynomial},
sponsor = {FWF project P16357-N04},
length = {18},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}

@techreport{RISC2464,
author = {Hong Gu and Martin Burger},
title = {{Preprocessing for Finite Element Discretizations of Geometric Problems}},
language = {english},
abstract = {In this paper, we use finite element method to approximate thesolutions of parameter-dependent geometric problems, andinvestigate the possibility of using symbolic methods as a preprocessing step.The main idea of our approach is to construct suitable finite elementdiscretizations of the nonlinear elliptic equations leading tosystems of algebraic equations, which can be subsequently solved bysymbolic computation within the tolerance of computer algebra software. The prolongation of the preprocessed symbolicsolution can serve as a starting value for a numerical iterative method ona finer grid.A motivation for this approach is that usual numerical iterations (e.g. viaNewton-type or fixed-point iterations) may diverge if no appropriate initialvalues are available.Moreover, such a purely numerical approach will not find all solutions of thediscretized problem if there are more than one. A final motivation for the use ofsymbolic methodsis the fact that all discrete solutions can be obtained as functions of unknownparameters. In this paper, we focus on a special class ofpartial differential equations derived from geometric problems. A mainchallenge in this class is the fact that the polynomial structure of thenonlinearity is not explicit in thedivergence form usually used for finite element discretization. As a consequence,the discrete form would always yield some non-polynomialterms. We therefore consider two different discretizations, namely a polynomialreformulation before discretization and a direct discretization of the divergence form withpolynomial approximation of the discrete system.In order to perform a detailed analysis and convergence theory of the discretizationmethods we investigate some model problems related to mean-curvaturetype equations.},
number = {05-02},
year = {2005},
month = {May},
note = {accepted to be published in proceeding of SNC'05},
keywords = {Finite element methods, symbolic computation, preprocessing, Newton iteration, multigrid, polynomial equations},
sponsor = {SFB F013/F1304, F1308},
length = {19},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}

@techreport{RISC2471,
author = {F.Winkler and M.Zhou},
title = {{On computing Groebner bases in rings of differential operators with coefficients in a ring}},
language = {english},
number = {05-04},
year = {2005},
sponsor = {This work has been supported by the FWF project P16357-N04.},
length = {13},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}

@techreport{RISC2478,
author = {M. Borges-Quintana and M. A. Borges-Trenard and E. Martinez-Moro and F. Winkler},
title = {{Using Groebner bases for determining the equivalence of linear codes and solving the decoding problem}},
language = {english},
number = {05-12},
year = {2005},
length = {32},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}