# Computer Algebra Tools for Special Functions [DK6]

### Project Lead

### Project Duration

01/10/2011 - 31/12/2020### Project URL

Go to Website## Partners

### The Austrian Science Fund (FWF)

## Publications

### 2018

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

#### Ralf Hemmecke, Silviu Radu

Technical report no. 18-03 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. January 26 2018. Accepted for publication in the Journal of Symbolic Computation. [pdf]@

author = {Ralf Hemmecke and Silviu Radu},

title = {{Construction of all Polynomial Relations among Dedekind Eta Functions of Level $N$}},

language = {english},

abstract = {We describe an algorithm that, given a positive integer $N$,computes a Gr\"obner 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 aGr\"obner 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)], \quad E_k\mapsto \eta(\delta_k\tau), \quad k=1,\ldots,n.\end{gather*}},

number = {18-03},

year = {2018},

month = {January 26},

note = {Accepted for publication in the Journal of Symbolic Computation},

keywords = {Dedekind $\eta$ function, modular functions, modular equations, ideal of relations, Groebner basis},

length = {18},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

**techreport**{RISC5561,author = {Ralf Hemmecke and Silviu Radu},

title = {{Construction of all Polynomial Relations among Dedekind Eta Functions of Level $N$}},

language = {english},

abstract = {We describe an algorithm that, given a positive integer $N$,computes a Gr\"obner 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 aGr\"obner 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)], \quad E_k\mapsto \eta(\delta_k\tau), \quad k=1,\ldots,n.\end{gather*}},

number = {18-03},

year = {2018},

month = {January 26},

note = {Accepted for publication in the Journal of Symbolic Computation},

keywords = {Dedekind $\eta$ function, modular functions, modular equations, ideal of relations, Groebner basis},

length = {18},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### The Generators of all Polynomial Relations among Jacobi Theta Functions

#### Ralf Hemmecke, Silviu Radu, Liangjie Ye

Technical report no. 18-09 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. July 9 2018. [pdf]@

author = {Ralf Hemmecke and Silviu Radu and Liangjie Ye},

title = {{The Generators of all Polynomial Relations among Jacobi Theta Functions}},

language = {english},

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 polynomialrelations among them with coefficients in$K :=\setQ(\theta_2(0|\tau),\theta_3(0|\tau),\theta_4(0|\tau))$ isgenerated by just two polynomials, that correspond to well knownidentities among Jacobi theta functions.},

number = {18-09},

year = {2018},

month = {July 9},

length = {9},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

**techreport**{RISC5719,author = {Ralf Hemmecke and Silviu Radu and Liangjie Ye},

title = {{The Generators of all Polynomial Relations among Jacobi Theta Functions}},

language = {english},

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 polynomialrelations among them with coefficients in$K :=\setQ(\theta_2(0|\tau),\theta_3(0|\tau),\theta_4(0|\tau))$ isgenerated by just two polynomials, that correspond to well knownidentities among Jacobi theta functions.},

number = {18-09},

year = {2018},

month = {July 9},

length = {9},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### 2017

### Elliptic Function Based Algorithms to Prove Jacobi Theta Function Relations

#### Liangjie Ye

Journal of Symbolic Computation, to appear, pp. 1-25. 2017. -. [pdf]@

author = {Liangjie Ye},

title = {{Elliptic Function Based Algorithms to Prove Jacobi Theta Function Relations}},

language = {english},

journal = {Journal of Symbolic Computation, to appear},

pages = {1--25},

isbn_issn = {-},

year = {2017},

refereed = {yes},

length = {25}

}

**article**{RISC5286,author = {Liangjie Ye},

title = {{Elliptic Function Based Algorithms to Prove Jacobi Theta Function Relations}},

language = {english},

journal = {Journal of Symbolic Computation, to appear},

pages = {1--25},

isbn_issn = {-},

year = {2017},

refereed = {yes},

length = {25}

}

### A Symbolic Decision Procedure for Relations Arising among Taylor Coefficients of Classical Jacobi Theta Functions

#### Liangjie Ye

Journal of Symbolic Computation 82, pp. 134-163. 2017. ISSN: 0747-7171. [pdf]@

author = {Liangjie Ye},

title = {{A Symbolic Decision Procedure for Relations Arising among Taylor Coefficients of Classical Jacobi Theta Functions}},

language = {english},

journal = {Journal of Symbolic Computation },

volume = {82},

pages = {134--163},

isbn_issn = {ISSN: 0747-7171},

year = {2017},

refereed = {yes},

length = {30}

}

**article**{RISC5455,author = {Liangjie Ye},

title = {{A Symbolic Decision Procedure for Relations Arising among Taylor Coefficients of Classical Jacobi Theta Functions}},

language = {english},

journal = {Journal of Symbolic Computation },

volume = {82},

pages = {134--163},

isbn_issn = {ISSN: 0747-7171},

year = {2017},

refereed = {yes},

length = {30}

}

### Complex Analysis Based Computer Algebra Algorithms for Proving Jacobi Theta Function Identities

#### Liangjie Ye

RISC and the DK program Linz. PhD Thesis. 2017. Updated version in June 2017. [pdf]@

author = {Liangjie Ye},

title = {{Complex Analysis Based Computer Algebra Algorithms for Proving Jacobi Theta Function Identities}},

language = {english},

year = {2017},

note = {Updated version in June 2017},

translation = {0},

school = {RISC and the DK program Linz},

length = {122}

}

**phdthesis**{RISC5463,author = {Liangjie Ye},

title = {{Complex Analysis Based Computer Algebra Algorithms for Proving Jacobi Theta Function Identities}},

language = {english},

year = {2017},

note = {Updated version in June 2017},

translation = {0},

school = {RISC and the DK program Linz},

length = {122}

}

### 2015

### Polyhedral Omega: A New Algorithm for Solving Linear Diophantine Systems

#### Felix Breuer, Zafeirakis Zafeirakopoulos

Technical report no. 15-09 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. January 2015. [pdf]@

author = {Felix Breuer and Zafeirakis Zafeirakopoulos},

title = {{Polyhedral Omega: A New Algorithm for Solving Linear Diophantine Systems}},

language = {english},

abstract = {Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon's iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decompositions and Barvinok's short rational function representations. In this way, we connect two recent branches of research that have so far remained separate, unified by the concept of symbolic cones which we introduce. The resulting LDS solver Polyhedral Omega is significantly faster than previous solvers based on partition analysis and it is competitive with state-of-the-art LDS solvers based on geometric methods. Most importantly, this synthesis of ideas makes Polyhedral Omega the simplest algorithm for solving linear Diophantine systems available to date. Moreover, we provide an illustrated geometric interpretation of partition analysis, with the aim of making ideas from both areas accessible to readers from a wide range of backgrounds.},

number = {15-09},

year = {2015},

month = {January},

keywords = {Linear Diophantine system, linear inequality system, integer solutions, partition analysis, partition theory, polyhedral geometry, rational function, symbolic cone, generating function, implementation},

length = {49},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

**techreport**{RISC5153,author = {Felix Breuer and Zafeirakis Zafeirakopoulos},

title = {{Polyhedral Omega: A New Algorithm for Solving Linear Diophantine Systems}},

language = {english},

abstract = {Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon's iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decompositions and Barvinok's short rational function representations. In this way, we connect two recent branches of research that have so far remained separate, unified by the concept of symbolic cones which we introduce. The resulting LDS solver Polyhedral Omega is significantly faster than previous solvers based on partition analysis and it is competitive with state-of-the-art LDS solvers based on geometric methods. Most importantly, this synthesis of ideas makes Polyhedral Omega the simplest algorithm for solving linear Diophantine systems available to date. Moreover, we provide an illustrated geometric interpretation of partition analysis, with the aim of making ideas from both areas accessible to readers from a wide range of backgrounds.},

number = {15-09},

year = {2015},

month = {January},

keywords = {Linear Diophantine system, linear inequality system, integer solutions, partition analysis, partition theory, polyhedral geometry, rational function, symbolic cone, generating function, implementation},

length = {49},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### 2011

### Lower Bounds and Constructions for q-ary Codes Correcting Asymmetric Errors

#### Qunying Liao, Liangjie Ye

Advances in Mathematics(China) 42(6), pp. 795-800. 2011. ISSN:1000-0917 . [url] [pdf]@

author = {Qunying Liao and Liangjie Ye},

title = {{Lower Bounds and Constructions for q-ary Codes Correcting Asymmetric Errors}},

language = {english},

journal = {Advances in Mathematics(China)},

volume = {42},

number = {6},

pages = {795--800},

isbn_issn = {ISSN:1000-0917 },

year = {2011},

refereed = {yes},

length = {6},

url = {http://advmath.pku.edu.cn/EN/volumn/volumn_1337.shtml}

}

**article**{RISC4569,author = {Qunying Liao and Liangjie Ye},

title = {{Lower Bounds and Constructions for q-ary Codes Correcting Asymmetric Errors}},

language = {english},

journal = {Advances in Mathematics(China)},

volume = {42},

number = {6},

pages = {795--800},

isbn_issn = {ISSN:1000-0917 },

year = {2011},

refereed = {yes},

length = {6},

url = {http://advmath.pku.edu.cn/EN/volumn/volumn_1337.shtml}

}