# Publications

Most of my papers are available online as technical reports.

*Articles*

- Ralf Hemmecke, Peter Paule, Silviu Radu:
**Construction of modular function bases for $\Gamma_0(121)$**related to $p(11n+6)$,*Integral Transforms and Special Functions*32(5-8):512-527, 2021. https://doi.org/10.1080/10652469.2020.1806261

Further Information - Ralf Hemmecke, Silviu Radu, Liangjie Ye:
**The Generators of all Polynomial Relations among Jacobi Theta Functions**, pp. 259-268, in: J. Blümlein, C. Schneider, Peter Paule (eds,) Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, Springer, 2019. https://doi.org/10.1007/978-3-030-04480-0_11 - Ralf Hemmecke and Silviu Radu.
**Construction of all Polynomial Relations among Dedekind Eta Functions of Level N**.*Journal of Symbolic Computation*, 95:39-52, 2019. https://doi.org/10.1016/j.jsc.2018.10.001 - Ralf Hemmecke.
**Dancing samba with Ramanujan partition congruences**.*Journal of Symbolic Computation*, 84:14-24, 2018. https://doi.org/10.1016/j.jsc.2017.02.001 - Joachim Apel and Ralf Hemmecke.
**Detecting unnecessary reductions in an involutive basis computation**.*Journal of Symbolic Computation*, 40(4-5):1131-1149, 2005. https://doi.org/10.1016/j.jsc.2004.04.004 - Erik Hillgarter, Ralf Hemmecke, Günter Landsmann, and Franz Winkler.
**Symbolic differential elimination theory for symmetry analysis**.*Mathematical and Computer Modelling of Dynamical Systems*, 10(2):123-147, June 2004. https://doi.org/10.1080/13873950412331318107 - Ralf Hemmecke.
**Dynamical aspects of involutive bases computations**. In Franz Winkler and Ulrich Langer, editors,*Symbolic and Numeric Scientific Computation*, number 2630 in Lecture Notes in Computer Science, pages 168-182. Springer-Verlag, 2003. https://link.springer.com/book/10.1007/3-540-45084-X - Ralf Hemmecke.
**Continuously parameterized symmetries and Buchberger's algorithm**.*Journal of Symbolic Computation*, 33(1):43-55, January 2002. https://doi.org/10.1006/jsco.2001.0478 - Ralf Hemmecke, Erik Hillgarter, and Franz Winkler. CASA. In J. Grabmeier, E. Kaltofen, and V. Weispfenning, editors,
*Computer Algebra Handbook: Foundation, Applications, Systems*, chapter 4.2.8, pages 356-359. Springer-Verlag, Heidelberg, 2003. https://www.springer.com/us/book/9783540654667

*Talks*

- Ralf Hemmecke. Calix--An Aldor package to compute Gröbner and involutive bases, November 2003. Talk presented at INRIA, Sophia-Antipolis, France.
- Ralf Hemmecke. Involutive divisions and involutive bases in the polynomial case, September 2003. Talk presented at MSRI, Berkeley, California.
- Ralf Hemmecke. Detecting unnecessary reductions in an involutive basis computation, June 2002. Talk presented at the Conference on Applications of Computer Algebra (ACA'02), Volos, Greece. Joint work with J. Apel.
- Ralf Hemmecke. A contribution to the faster computation of involutive bases, May 2002. Talk presented at the University of Leipzig, Germany.
- Ralf Hemmecke. An efficient method for finding a simplifier in an involutive basis computation. In Jacques Calmet, Marcus Hausdorf, and Werner M. Seiler, editors,
*Proceedings of the Workshop on Under- and Overdetermined Systems of Algebraic or Differential Equations, March 18-19, 2002, Karlsruhe, Germany*, pages 87-95. Institut für Algorithmen und Kognitive Systeme, Universität Karlsruhe, March 2002. Local proceedings. - Ralf Hemmecke. Dynamical aspects of involutive bases computations, September 2001. Talk presented at the Conference on Symbolic and Numeric Scientific Computation (SNSC'01), Hagenberg, Austria.
- Ralf Hemmecke. Dynamical aspects of involutive bases computations, June 2001. Talk presented at the Conference on Applications of Computer Algebra (ACA'01), Albuquerque, New Mexico.
- Ralf Hemmecke. Computational algebraic geometry with CASA, April 2001. Talk presented at a German SFB Workshop in Herrsching/Ammersee (Germany).
- Ralf Hemmecke. CASA--A Maple package to investigate algebraic curves, December 1999. Talk presented at the University of Duisburg, Germany.

*Diploma and PhD Theses*

- Ralf Hemmecke. Lösen von Gleichungssystemen mit kontinuierlichen Symmetrien. Diplomarbeit, Universität Leipzig, Augustusplatz 10-11, 04109 Leipzig, Germany, February 1996.
- Ralf Hemmecke.
*Involutive Bases for Polynomial Ideals*. PhD thesis, Universität Linz, 4040 Linz, Austria, February 2003.

- Ralf Hemmecke, Peter Paule, Silviu Radu:

Construction of an Integral Basis for $M^\infty(121)$,

RISC report 19-10

also available as: https://doi.org/10.1080/10652469.2020.1806261*Abstract:*Motivated by arithmetic properties of partition numbers $p(n)$, our goal is to find algorithmically a Ramanujan type identity of the form $\sum_{n=0}^{\infty}p(11n+6)q^n=R$, where $R$ is a polynomial in products of the form $e_\alpha:=\prod_{n=1}^{\infty}(1-q^{11^\alpha n})$ with $\alpha=0,1,2$. To this end we multiply the left side by an appropriate factor such the result is a modular function for $\Gamma_0(121)$ having only poles at infinity. It turns out that polynomials in the $e_\alpha$ do not generate the full space of such functions, so we were led to modify our goal. More concretely, we give three different ways to construct the space of modular functions for $\Gamma_0(121)$ having only poles at infinity. This in turn leads to three different representations of $R$ not solely in terms of the $e_\alpha$ but, for example, by using as generators also other functions like the modular invariant $j$.

Further Information - Ralf Hemmecke, Silviu Radu, Liangjie Ye:

The Generators of all Polynomial Relations among Jacobi Theta Functions

RISC report 18-09

also available as: https://doi.org/10.1007/978-3-030-04480-0_11*Abstract:*

In this article, we consider the classical Jacobi theta functions $\theta_i(z)$, $i=1,2,3,4$ and show that the ideal of all polynomial relations among them with coefficients in $K :=Q(\theta_2(0|\tau),\theta_3(0|\tau),\theta_4(0|\tau))$ is generated by just two polynomials, that correspond to well known identities among Jacobi theta functions . - Ralf Hemmecke and Silviu Radu:

Construction of all Polynomial Relations among Dedekind Eta Functions of Level N

RISC report 18-03

also available as https://doi.org/10.1016/j.jsc.2018.10.001*Abstract:*We describe an algorithm that, given a positive integer $N$, computes a Gröbner basis of the ideal of polynomial relations among Dedekind $\eta$-functions of level $N$, i.e., among the elements of $\{\eta(\delta_1\tau),\ldots,\eta(\delta_n\tau)\}$ where $1=\delta_1<\delta_2\dots>\delta_n=N$ are the positive divisors of $N$. More precisely, we find a finite generating set (which is also a Gröbner basis) of the ideal $\ker\phi$ where

\begin{gather*}

\phi:Q[E_1,\ldots,E_n] \to Q[\eta(\delta_1\tau),\ldots,\eta(\delta_n\tau)],

E_k\mapsto \eta(\delta_k\tau),

k=1,\ldots,n.

\end{gather*} - Ralf Hemmecke:

Dancing samba with Ramanujan partition congruences

RISC report 16-06

also available as https://doi.org/10.1016/j.jsc.2017.02.001*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 $n$. - Ralf Hemmecke, Erik Hillgarter, Günter Landsmann, and Franz Winkler:

Symbolic Differential Elimination for Symmetry Analysis

SFB F013 Report 2003-08.*Abstract:*Differential problems are ubiquitous in mathematical modeling of physical and scientific problems. Algebraic analysis of differential systems can help in determining qualitative and quantitative properties of solutions of such systems. We describe several algebraic methods for investigating differential systems. - Ralf Hemmecke:

Involutive Bases for Polynomial Ideals

PhD Thesis, Johannes Kepler Universität Linz, 2003

RISC report 03-02*Abstract:*This thesis contributes to the theory of polynomial involutive bases. Firstly, we present the two existing theories of involutive divisions, compare them, and come up with a generalised approach of*suitable partial divisions*. The thesis is built on this generalised approach. Secondly, we treat the question of choosing a "good" suitable partial division in each iteration of the involutive basis algorithm. We devise an efficient and flexible algorithm for this purpose, the*Sliced Division*algorithm. During the involutive basis algorithm, the Sliced Division algorithm contributes to an early detection of the involutive basis property and a minimisation of the number of critical elements. Thirdly, we give new criteria to avoid unnecessary reductions in an involutive basis algorithm. We show that the termination property of an involutive basis algorithm which applies our criteria is independent of the prolongation selection strategy used during its run. Finally, we present an implementation of the algorithms and results of this thesis in our software package CALIX. - Joachim Apel and Ralf Hemmecke:

Detecting Unnecessary Reductions in an Involutive Basis Computation

RISC report 02-22

also available as https://doi.org/10.1016/j.jsc.2004.04.004

*Abstract:*We consider the check of the involutive basis property in a polynomial context. In order to show that a finite generating set*F*of a polynomial ideal*I*is an involutive basis one must confirm two properties. Firstly, the set of leading terms of the elements of*F*has to be complete. Secondly, one has to prove that*F*is a Gröbner basis of*I*. The latter is the time critical part but can be accelerated by application of Buchberger's criteria including the many improvements found during the last two decades.Gerdt and Blinkov (Involutive Bases of Polynomial Ideals.

*Mathematics and Computers in Simulation***45**, pp. 519-541, 1998) were the first who applied these criteria in involutive basis computations. We present criteria which are also transfered from the theory of Gröbner bases to involutive basis computations. We illustrate that our results exploit the Gröbner basis theory slightely more than those of Gerdt and Blinkov. Our criteria apply in all cases where those of Gerdt/Blinkov do, but we also present examples where our criteria are superior.Some of our criteria can be used also in algebras of solvable type, e.g., Weyl algebras or enveloping algebras of Lie algebras, in full analogy to the Gröbner basis case.

We show that the application of criteria enforces the termination of the involutive basis algorithm independent of the prolongation selection strategy.

- Ralf Hemmecke:

Dynamical Aspects of Involutive Bases Computations

RISC report 01-28

also available from Springer*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.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.

- Ralf Hemmecke, Erik Hillgarter, Wolfgang Schreiner, Franz Winkler:

An Evaluation of the State of the CASA System

RISC report 98-16

Research Institute for Symbolic Computation, Johannes Kepler Universität, 4040 Linz, Austria, Europe, October 1998. - Ralf Hemmecke:

Lösen von Gleichungssystemen mit kontinuierlichen Symmetrien

Diplomarbeit, Universität Leipzig, 1996. - Ralf Hemmecke:

Investigations about Monads

University of Kent, Canterbury, 1993.