Research

List of papers sorted by topics. Some papers are listed several times as they cover more than one topic. The list of papers in reverse chronological order is found in the CV and on my Google Scholar profile. A summary of my research highlights is given in my research statement.

Elimination theory

• The Newton-Puiseux algorithm and effective algebraic series, Journal of Algebra, 688, 284-306, 2026 [link]
  _{We explain how to encode an algebraic series by finite data and how to do effective arithmetic on the level of these encodings. The reasoning is based on the Newton-Puiseux algorithm and an effective equality test for algebraic series. We also discuss how to derive information about the support of an algebraic series. Based thereon, we show how to identify the polynomial and rational solutions of a polynomial equation.}

• The orbit-sum method for higher order equations, with Manuel Kauers, to appear in Annals of Combinatorics, 2022 [arXiv]
  _{The orbit-sum method is an algebraic version of the reflection-principle that was introduced by Bousquet-Mélou and Mishna to solve functional equations that arise in the enumeration of lattice walks with small steps restricted to $\mathbb{N}^2$. It proceeds by computing a set of algebraic substitutions that can be applied to a given functional equation, forming a linear combination of its transformed versions to the end of eliminating some of the unknowns, and eliminating further unknowns by discarding terms with negative powers. The extension of the orbit-sum method to walks with large steps was started by Bostan, Bousquet-Mélou and Melczer. They presented an algorithm that computes the minimal polynomials of the algebraic substitutions. We continue their work by explaining, among other things, how to perform computations in their splitting field on the level of ``formal’’ algebraic extensions and how its elements can be interpreted as series. We thereby make use of the primitive element theorem, Gröbner bases and the shape lemma, and the Newton-Puiseux algorithm.}

• Separating variables in bivariate polynomial ideals, with Manuel Kauers and Gleb Pogudin, Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation, 54 - 61, 2020 [link]
  _{We present an algorithm which for any given ideal $I\subseteq\mathbb{K}[x,y]$ finds all elements of $I$ that have the form $f(x) − g(y)$, that is, all elements in which no monomial is a multiple of $xy$.}

• Separating variables in bivariate polynomial ideals: the local case, submitted, 2024 [arXiv]
  _{We present a semi-algorithm which for any irreducible polynomial $p\in\mathbb{K}[x,y]$ finds all elements of $\mathbb{K}(x) + \mathbb{K}(y)$ whose numerator is a multiple of $p$.}

• Separated variables on plane algebraic curves, submitted, 2024 [arXiv]
  _{We investigate the problem of deciding whether the restriction of a rational function $r\in\mathbb{K}(x,y)$ to the curve associated with an irreducible polynomial $p\in\mathbb{K}[x,y]$ is the restriction of an element of $\mathbb{K}(x)+\mathbb{K}(y)$. We present an algorithm and a conjectural semi-algorithm for finding such elements depending on whether $p$ has a non-trivial rational multiple in $\mathbb{K}(x) + \mathbb{K}(y)$ or not.}

• On finite orbits of infinite correspondences, 2025 [arXiv]
  _{We collect results about algebraic correspondences and adapt them to the setting of correspondences on projective lines. The focus lies on finite orbits of algebraic correspondences. The main result is a field theoretic characterization of the (in)finiteness of the number of finite orbits.}

• On the problem of separating variables in multivariate polynomial ideals, with Manuel Kauers, Proceedings of the 49th International Symposium on Symbolic and Algebraic Computation, 100 - 107, 2024 [link]
  _{For a given ideal $I \subseteq \mathbb{K}[x_1,\dots,x_n,y_1,\dots,y_m]$ in a polynomial ring with $n+m$ variables, we want to find all elements that can be written as $f - g$ for some $f \in \mathbb{K}[x_1,\dots,x_n]$ and some $g\in\mathbb{K}\in \mathbb{K}[y_1,\dots,y_m]$, i.e., all elements of $I$ that contain no term involving at the same time one of the $x_1,\dots,x_n$ and one of the $y_1,\dots,y_m$. For principal ideals and for ideals of dimension zero, we give algorithms that compute all these polynomials in a finite number of steps.}

Dynamical systems

• On finite orbits of infinite correspondences, 2025 [arXiv]
  _{We collect results about algebraic correspondences and adapt them to the setting of correspondences on projective lines. The focus lies on finite orbits of algebraic correspondences. The main result is a field theoretic characterization of the (in)finiteness of the number of finite orbits.}

Enumeration of lattice walks

• $x(1-t(x+x^{-1}) F(x;t) = x - tF(0;t),$ submitted, 2025 [arXiv]
  _{The purpose of these notes is to introduce some of the problems the enumeration of lattice walks is dedicated to and familiarize with some of the arguments they can be addressed with. We discuss the enumeration of lattice walks, their generating functions, and the functional equations they satisfy. We focus on algebraic methods for manipulating and solving these equations. Elementary power series algebra plays a prominent role, computer algebra too, but we repeatedly digress and present ideas and methods of different kind whenever it seems appropriate. The exposition is organized around the most simple yet non-trivial problem in the enumeration of lattice walks. The intention is to illustrate different techniques without getting technical.}

• The orbit-sum method for higher order equations, with Manuel Kauers, to appear in Annals of Combinatorics, 2022 [arXiv]
  _{The orbit-sum method is an algebraic version of the reflection-principle that was introduced by Bousquet-Mélou and Mishna to solve functional equations that arise in the enumeration of lattice walks with small steps restricted to $\mathbb{N}^2$. It proceeds by computing a set of algebraic substitutions that can be applied to a given functional equation, forming a linear combination of its transformed versions to the end of eliminating some of the unknowns, and eliminating further unknowns by discarding terms with negative powers. The extension of the orbit-sum method to walks with large steps was started by Bostan, Bousquet-Mélou and Melczer. They presented an algorithm that computes the minimal polynomials of the algebraic substitutions. We continue their work by explaining, among other things, how to perform computations in their splitting field on the level of ``formal’’ algebraic extensions and how its elements can be interpreted as series. We thereby make use of the primitive element theorem, Gröbner bases and the shape lemma, and the Newton-Puiseux algorithm.}

• The Newton-Puiseux algorithm and effective algebraic series, Journal of Algebra, 688, 284-306, 2026 [link]
  _{We explain how to encode an algebraic series by finite data and how to do effective arithmetic on the level of these encodings. The reasoning is based on the Newton-Puiseux algorithm and an effective equality test for algebraic series. We also discuss how to derive information about the support of an algebraic series. Based thereon, we show how to identify the polynomial and rational solutions of a polynomial equation.}

• Inhomogeneous restricted lattice walks, with Manuel Kauers, Séminaire Lotharingien de Combinatoire, 82B, Art. 75, 12pp, 2019 [link]
  _{We consider inhomogeneous lattice walk models in a half-space and in the quarter plane. For the models in a half-space, we show by a generalization of the kernel method to linear systems of functional equations that their generating functions are always algebraic. For the models in the quarter plane, we have carried out an experimental classification of all models with small steps. We discovered many (apparently) D-finite cases for most of which we have no explanation yet.}

• Quadrant walks starting outside the quadrant, with Manuel Kauers and Amélie Trotignon, Séminaire Lotharingien de Combinatoire, 75B, Art. 26, 11pp, 2021 [link]
  _{We investigate a functional equation which resembles the functional equation for the generating function of a lattice walk model for the quarter plane. The interesting feature of this equation is that its orbit sum is zero while its solution is not algebraic. The solution can be interpreted as the generating function of lattice walks in $\mathbb{Z}^2$ starting at $(-1,-1)$ and subject to the restriction that the coordinate axes can be crossed only in one direction. We also consider certain variants of the equation, all of which seem to have transcendental solutions. In one case, the solution is perhaps not even D-finite.}

• Walks with small steps in the 4D-orthant, with Sophie Hofmanninger and Manuel Kauers, Annals of Combinatorics, 25, 153–166, 2021 [link]
  _{We provide some first experimental data about generating functions of restricted lattice walks with small steps in $\mathbb{N}^4$.}

Optimization

• Graph learning based on total variation minimization, with Peter Berger, Gabor Hannak and Gerald Matz,Proceedings of the 2018 IEEE International Conference on Acoustics, Speech and Signal Processing, 6309-6313, 2018 [link]
  _{We consider the problem of learning the topology of a graph from a given set of smooth graph signals. We construct a weighted adjacency matrix that best explains the data in the sense of achieving the smallest graph total variation. For the case of noisy measurements of the graph signals we propose a scheme that simultaneously denoises the signals and learns the graph adjacency matrix. Our method allows for a direct control of the number of edges and of the weighted node degree. Numerical experiments demonstrate that our graph learning scheme is well suited for community detection.}

Theses

• Algorithms for the enumeration of lattice walks, PhD thesis, Johannes Kepler University Linz, 2021 [link]
  _{The dissertation (almost literally) includes 3 of the published papers: Inhomogeneous restricted lattice walks, Separating variables in bivariate polynomial ideals, and Quadrant walks starting outside the quadrant. Each of them constitutes a chapter, and each of these chapters comes with an extended introduction or appendix which carefully motivates the problem or in detail explains the underlying computations. Much of the paper $x(1-(x+x^{-1}))F(x;t) = x - F(0;t)$ is found in the chapter on preliminaries, and many of the ideas that underly The orbit-sum method for higher-order equations and The Newton-Puiseux algorithm and effective algebraic series are discussed in the chapter on time-inhomogeneous lattice walks. The classification of time-inhomogeneous lattice walks in the quarter plane therein has not been published elsewhere and can only be found in the thesis.}

• The asymptotic behaviour of denumerable Markov chains, Master’s thesis, University of Vienna, 2015 [link]