# Dipl.-Math. Dr. Ralf Hemmecke

Computer Algebra

### Office

Schloss Hagenberg
A-4232 Hagenberg im Mühlkreis

Room: 2.7-1

Research Institute for Symbolic Computation
Johannes Kepler University
Altenberger Straße 69
A-4040 Linz, Austria

### eMail

Ralf.Hemmecke@risc.jku.at

## Courses

Course Id Title Registration Type Hours Teachers Rhythm
 Content provided by KUSSS, Johannes Kepler University Linz | E-Mail

## Publications

### Dancing Samba with Ramanujan Partition Congruences

#### Ralf Hemmecke

Journal of Symbolic Compuation 84, pp. 14-24. 2018. ISSN 0747-7171. [url]
@article{RISC5703,
author = {Ralf Hemmecke},
title = {{Dancing Samba with Ramanujan Partition Congruences}},
language = {english},
abstract = {The article presents an algorithm to compute a $C[t]$-module basis $G$ for a given subalgebra $A$ over a polynomial ring $R=C[x]$ with a Euclidean domain $C$ as the domain of coefficients and $t$ a given element of $A$. The reduction modulo $G$ allows a subalgebra membership test. The algorithm also works for more general rings $R$, in particular for a ring $R\subset C((q))$ with the property that $f\in R$ is zero if and only if the order of $f$ is positive. As an application, we algorithmically derive an explicit identity (in terms of quotients of Dedekind $\eta$-functions and Klein's $j$-invariant) that shows that $p(11n+6)$ is divisible by 11 for every natural number $n$ where $p(n)$ denotes the number of partitions of $n$.},
journal = {Journal of Symbolic Compuation},
volume = {84},
pages = {14--24},
isbn_issn = {ISSN 0747-7171},
year = {2018},
refereed = {yes},
keywords = {Partition identities, Number theoretic algorithm, Subalgebra basis},
length = {11},
url = {http://www.sciencedirect.com/science/article/pii/S0747717117300147}
}

### Detecting Unnecessary Reductions in an Involutive Basis Computation

#### Joachim Apel, Ralf Hemmecke

Journal of Symbolic Computation 40(4-5), pp. 1131-1149. 2005. ISSN 0747-7171. [url]
@article{RISC2492,
author = {Joachim Apel and Ralf Hemmecke},
title = {{Detecting Unnecessary Reductions in an Involutive Basis Computation}},
language = {english},
abstract = {We consider the check of the involutive basis property in a polynomialcontext. In order to show that a finite generating set $F$ of apolynomial ideal $I$ is an involutive basis one must confirm twoproperties. Firstly, the set of leading terms of the elements of $F$has to be complete. Secondly, one has to prove that $F$ is a Gr�bnerbasis of $I$. The latter is the time critical part but can beaccelerated by application of Buchberger's criteria including the manyimprovements found during the last two decades.Gerdt and Blinkov (Involutive Bases of Polynomial Ideals. {\em Mathematics and Computers in Simulation} {\bf 45}, pp.~519--541,1998) were the first who applied these criteria in involutive basiscomputations.We present criteria which are also transferred from the theory ofGr�bner bases to involutive basis computations. We illustrate that ourresults exploit the Gr�bner basis theory slightly more than those ofGerdt and Blinkov. Our criteria apply in all cases where those ofGerdt/Blinkov do, but we also present examples where our criteria aresuperior. Some of our criteria can be used also in algebras of solvable type,\eg, Weyl algebras or enveloping algebras of Lie algebras, in fullanalogy to the Gr�bner basis case.We show that the application of criteria enforces the termination ofthe involutive basis algorithm independent of the prolongationselection strategy.},
journal = {Journal of Symbolic Computation},
volume = {40},
number = {4--5},
pages = {1131--1149},
isbn_issn = {ISSN 0747-7171},
year = {2005},
refereed = {yes},
sponsor = {FWF SFB F013, project 1304; Naturwissenschaftlich-Theoretisches Zentrum (NTZ) of the University of Leipzig, Germany},
length = {19},
url = {http://www.sciencedirect.com/science/journal/07477171}
}

### Symbolic Differential Elimination for Symmetry Analysis

#### Erik Hillgarter, Ralf Hemmecke, Günter Landsmann, Franz Winkler

Mathematical and Computer Modelling of Dynamical Systems 10(2), pp. 123-147. 2004. ISSN 1387-3954. [url] [pdf]
@article{RISC2295,
author = {Erik Hillgarter and Ralf Hemmecke and Günter Landsmann and Franz Winkler},
title = {{Symbolic Differential Elimination for Symmetry Analysis}},
language = {english},
journal = {Mathematical and Computer Modelling of Dynamical Systems},
volume = {10},
number = {2},
pages = {123--147},
isbn_issn = { ISSN 1387-3954},
year = {2004},
refereed = {yes},
length = {25},
url = {http://www.tandf.co.uk/journals/titles/13873954.asp}
}

### Symbolic Differential Elimination for Symmetry Analysis

#### R. Hemmecke, E. Hillgarter, G. Landsmann, F. Winkler

In: Proc. 4th IMACS Symposium on Mathematical Modelling, , pp. 790-796. Feb. 2003. Techn. Univ. Vienna, ISBN 3-901608-24-9. [pdf]
@inproceedings{RISC268,
author = {R. Hemmecke and E. Hillgarter and G. Landsmann and F. Winkler},
title = {{Symbolic Differential Elimination for Symmetry Analysis}},
booktitle = {{Proc. 4th IMACS Symposium on Mathematical Modelling}},
language = {english},
pages = {790--796},
isbn_issn = {ISBN 3-901608-24-9},
year = {2003},
month = {Feb.},
editor = {F. Breitenecker and I. Troch},
refereed = {yes},
institution = {Techn. Univ. Vienna},
length = {7}
}

### CASA

#### R. Hemmecke, E. Hillgarter, F. Winkler

In: Handbook of Computer Algebra: Foundations, Applications, Systems, , pp. 356-359. 2003. Springer-Verlag, ISBN 3-540-65466-6. [pdf]
@incollection{RISC269,
author = {R. Hemmecke and E. Hillgarter and F. Winkler},
title = {{CASA}},
booktitle = {{Handbook of Computer Algebra: Foundations, Applications, Systems}},
language = {english},
pages = {356--359},
publisher = {Springer-Verlag},
isbn_issn = {ISBN 3-540-65466-6},
year = {2003},
editor = {J. Grabmeier and E. Kaltofen and V. Weispfenning},
refereed = {yes},
sponsor = {Austrian FWF, SFB F013, project 1304},
length = {5}
}

### Dynamical Aspects of Involutive Bases Computations

#### Ralf Hemmecke

In: Symbolic and Numeric Scientific Computation, 2630, pp. 168-182. 2003. Springer Verlag, ISBN 3-540-40554-2. [url]
@inproceedings{RISC2293,
author = {Ralf Hemmecke},
title = {{Dynamical Aspects of Involutive Bases Computations}},
booktitle = {{Symbolic and Numeric Scientific Computation}},
language = {english},
abstract = {The article is a contribution to a more efficient computation of involutive bases. We present an algorithm which computes a sliced division'. A sliced division is an admissible partial division in the sense of Apel. Admissibility requires a certain order on the terms. Instead of ordering the terms in advance, our algorithm additionally returns such an order for which the computed sliced division is admissible. Our algorithm gives rise to a whole class of sliced divisions since there is some freedom to choose certain elements in the course of its run. We show that each sliced division refines the Thomas division and thus leads to terminating completion algorithms for the computation of involutive bases. A sliced division is such that its cones cover' a relatively `big' part of the term monoid generated by the given terms. The number of prolongations that must be considered during the involutive basis algorithm is tightly connected to the dimensions and number of the cones. By some computer experiments, we show how this new division can be fruitful for the involutive basis algorithm.\par We generalise the sliced division algorithm so that it can be seen as an algorithm which is parameterised by two choice functions and give particular choice functions for the computation of the classical divisions of Janet, Pommaret, and Thomas.},
number = {2630},
pages = {168--182},
publisher = {Springer Verlag},
isbn_issn = {ISBN 3-540-40554-2},
year = {2003},
editor = {Franz Winkler and Ulrich Langer},
refereed = {yes},
sponsor = {FWF, SFB F013, project 1304},
length = {15},
url = {http://www.springer.de/cgi/svcat/search_book.pl?isbn=3-540-40554-2}
}

### Involutive Bases for Polynomial Ideals

#### Ralf Hemmecke

Research Institute for Symbolic Computation. PhD Thesis. Johannes Kepler University, Linz, Austria, February 2003. [url]
@phdthesis{RISC3281,
author = {Ralf Hemmecke},
title = {{Involutive Bases for Polynomial Ideals}},
language = {english},
address = {Johannes Kepler University, Linz, Austria},
year = {2003},
month = {February},
translation = {0},
school = {Research Institute for Symbolic Computation},
keywords = {involutive basis, involutive division, suitable partial division},
sponsor = {FWF, SFB F013, project 1304},
length = {110},
}

### Continuously Parameterized Symmetries and Buchberger's Algorithm

#### Ralf Hemmecke

Journal of Symbolic Computation 33(1), pp. 43-55. January 2002. 0747-7171.
@article{RISC2290,
author = {Ralf Hemmecke},
title = {{Continuously Parameterized Symmetries and Buchberger's Algorithm}},
language = {english},
abstract = {Systems of polynomial equations often have symmetries. In solving such a system using Buchberger's algorithm, the symmetries are neglected. Incorporating symmetries into the solution process enables us to solve larger problems than with Buchberger's algorithm alone. This paper presents a method that shows how this can be achieved and also gives an algorithm that brings together continuously parameterized symmetries with Buchberger's algorithm.},
journal = {Journal of Symbolic Computation},
volume = {33},
number = {1},
pages = {43--55},
isbn_issn = {0747-7171},
year = {2002},
month = {January},
refereed = {yes},
length = {13}
}