## Research Interests

I work on **applied algebra**. My research covers a broad range of topics; my key interests are:

- Nonnegativity of real polynomials (real algebraic geometry).
- Polynomial optimization.
- Algebra related applications in science and engineering.
- Computational and algorithmic algebraic geometry, e.g., polynomial system solving.
- Mathematical software development.

There is a range of further topics I work on, am interested in, or have worked on in the past. For example:

- Amoeba theory (dealing with complex algebraic varieties under norm constraints).
- Fewnomial theory / sparsity of polynomial sysem.
- Combinatorial algebraic geometry (e.g., A-discriminants, toric, and tropical geometry).
- Various topics in optimization.

For me, applied Algebra means **applied** algebra: My group and I develop software, we work with real world data, and we cooperate with researchers from other sciences.
For example, we are involved in projects on

- chemical reaction networks,
- material design,
- life science,
- theoretical computer science.

## Publications

You can find my articles on the ArXiv.

### Research Articles

**"Nondegenerate Multistationarity in Small Reaction Networks"**

with A. Shiu; Discrete and Continuous Dynamical Systems, Series B,**24**(6) (June 2019), 2683-2700; see ArXiv 1802.00306.**"An Approach to Constrained Polynomial Optimization via Nonnegative Circuit Polynomials and Geometric Programming"**

with M. Dressler and S. Iliman; Journal of Symbolic Computation**91**, (2019), 149-172 (special issue for "MEGA 2017"); see ArXiv 1602.06180.**"Imaginary Projections of Polynomials"**

with T. Jörgens and T. Theobald,**91**, (2019), 181-199 Journal of Symbolic Computation (special issue for "MEGA 2017"); see ArXiv 1602.02008.

Was accepted for a talk at "MEGA 2017".**"Lopsided Approximation of Amoebas"**

with J. Forsgård, L. Matusevich, and N. Mehlhop,**8**(315) (2019), 485-500; see Mathematics of Computation; see ArXiv 1608.08663.**"A Positivstellensatz for Sums of Nonnegative Circuit Polynomials"**

with M. Dressler and S. Iliman; SIAM Journal on Applied Algebra and Geometry**1**(1) (2017), 536-555; see ArXiv.

Was accepted for a talk at "MEGA 2017".**"Lower Bounds for Polynomials with Simplex Newton Polytopes Based on Geometric Programming"**,

with S. Iliman; SIAM Journal on Optimization**26**(2) (2016), 1128-1146; see ArXiv.**"Amoebas, Nonnegative Polynomials and Sums of Squares Supported on Circuits"**,

with S. Iliman;, Research in the Mathematical Sciences**3**(1) (2016), 1-35; see ArXiv.

Was accepted for a talk at "MEGA 2015".**"Norms of Roots of Trinomials"**,

with T. Theobald; Mathematische Annalen,**366**(1) (2016), 219-247; ; see ArXiv.

Was accepted for a talk at "MEGA 2015".**"Separating Inequalities for Nonnegative Polynomials that Are not Sums of Squares"**,

with S. Iliman; Journal of Symbolic Computation**68**(2015), part 2, 181-194 (special issue for "MEGA 2013"); see ArXiv.

Was accepted for a talk at "MEGA 2013".**"Approximating Amoebas and Coamoebas by Sums of Squares"**,

with T. Theobald; Mathematics of Computation**84**(2015), 455-473; see ArXiv.

Was accepted for a poster presentation at "MEGA 2011".**"Amoebas of Genus at Most One"**,

with T. Theobald; Advances in Mathematics**239**(2013), 190-213; see ArXiv.

### Conference Publications

**"Computing the Real Isolated Points of an Algebraic Hypersurface"**

with P.L. Huu, and M. Safey El Din; Proceedings of ISSAC 2020, Kalamata, Greece, July 20-23, 2020 (virtual conference due to COVID-19) (I. Z. Emiris, and L. Zhi (eds.)), ACM, (2020), 297 - 304; see ArXiv 2008.10331.**"Global Optimization via the Dual SONC Cone and Linear Programming"**

with M. Dressler, H. Naumann, and J. Heuer; Proceedings of ISSAC 2020, Kalamata, Greece, July 20-23, 2020 (virtual conference due to COVID-19) (I. Z. Emiris, and L. Zhi (eds.)), ACM, (2020), 138 - 145; see ArXiv 2002.09368.**"New Dependencies of Hierarchies in Polynomial Optimization"**

with A. Kurpisz; Proceedings of ISSAC 2019, Beijing, China, July 15-18, 2019 (J. H. Davenport, D. Wang, M. Kauers, and R. J. Bradford (eds.)), ACM, (2019), 251-258; see ArXiv 1903.04996.**"Exact Optimization via Sums of Nonnegative Circuits and Sums of AM/GM Exponentials"**

with V. Magron and H. Seidler; Proceedings of ISSAC 2019, Beijing, China, July 15-18, 2019 (J. H. Davenport, D. Wang, M. Kauers, and R. J. Bradford (eds.)), ACM, (2019), 291-298; see ArXiv 1902.02123.**"A New Method for Computing Elimination Ideals of Likelihood Equations"**

with X. Tang and R. Zhao; Proceedings of ISSAC 2019, Beijing, China, July 15-18, 2019 (J. H. Davenport, D. Wang, M. Kauers, and R. J. Bradford (eds.)), ACM, (2019), 339-346; see ArXiv 1810.05620.

Suplimentary Material for the computations in the article are available via our website accompanying the article.**"Optimization over the Boolean Hypercube via Sums of Nonnegative Circuit Polynomials"**

with M. Dressler and A. Kurpisz; 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018, Liverpool, UK, August 27-31, 2018, (I. Potapov, P. G. Spirakis, and J. Worrell, eds.), LIPIcs, 117, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, (2018), 82:1-82:17; see ArXiv 1802.10004.

Was also accepted for a talk at "FLoC '18".

### Articles Accepted for Publication

**"Initial Steps in the Classification of Maximal Mediated Sets"**

with J. Hartzer, O. Röhrig, and O. Yürük; to appear in the Journal of Symbolic Computation (special issue for ``MEGA 2019''); published electronically on July 08, 2020; doi: 10.1016/j.jsc.2020.07.013; see ArXiv 1910.00502.**"The Kinetic Space of Multistationarity in Dual Phosphorylation"**

with E. Feliu, N. Kaihnsa, and O. Yürük; published electronically on Sep. 04, 2020; doi: 10.1007/s10884-020-09889-6; see ArXiv 2001.08285.

### Preprints

**"Evaluation of Pool-based Testing Approaches to Enable Population-wide Screening for COVID-19"**

with D.~Pflüger, M.~Rehme, J.~Heuer, and M.-I.~Bittner, see ArXiv 2004.11851.

This project has accompanying code and an accompanying interactive website.

News about the project at Die Junge Akademie, TU Braunschweig (in German), and Universität Stuttgart (in German).**"A Polyhedral Homotopy Algorithm For Real Zeros"**

with A. A. Ergür, see ArXiv 1910.01957.**"The Algebraic Boundary of the SONC Cone"**

with J. Forsgård, see ArXiv 1905.04776.**"An Experimental Comparison of SONC and SOS Certificates for Unconstrained Optimization"**

with H. Seidler, see ArXiv 1808.08431.

NOTE: The polynomial in Example 4.3 is not displayed correctly in the current Arxiv version. The correct polynomial is available via our website accompanying the article.

Was accepted for a talk at "MEGA 2019".**"The Lattice of Amoebas"**

with J. Forsgård, see ArXiv 1711.02705.**"Intersections of Amoebas"**,

with M. Juhnke-Kubitzke, see ArXiv 1510.08416.

Was accepted for a poster presentation at "FPSAC '16".**"A Sharp Upper Bound for the Complexity of Labeled Oriented Trees"**,

with M. Christmann, see ArXiv 1412.7257.**"The Boundary of Amoebas"**,

with F. Schroeter, see ArXiv 1310.7363.**"Low Dimensional Test Sets for Nonnegativity of Even Symmetric Forms"**,

with S. Iliman, see ArXiv 1303.4241.**"Polytopes with Special Simplices"**,

see ArXiv 1009.6158.

### Surveys

**"Amoebas, Nonnegative Polynomials and Sums of Squares Supported on Circuits"**,

an extended abstract for the correspondent article;

Oberwolfach Report, no. 23, 2015, 1308-1311; for the workshop "Tropical Aspects in Geometry, Topology and Physics".**"Amoebas and their Tropicalizations - a Survey"**, to appear in "Analysis Meets Geometry: The Mikael Passare Memorial Volume", M. Andersson, J. Boman, C. Kiselman, P. Kurasov, R. Sigurdsson (Eds.), Series: "Trends in Mathematics", Birkäuser Mathematics, 2017, 157-190.

### Theses

**"On the Geometry, Topology and Approximation of Amoebas"**,

PhD thesis for mathematics (Dissertation), 2013, download as PDF.**"Goodmans "New Riddle of Induction" - Eine Analyse auf mathematischer Grundlage"**,

Master thesis for philosophy (Magisterarbeit), 2010.**"Polytope mit speziellen Simplizes"**,

Master thesis for mathematics (Diplomarbeit), 2008, download as PDF (German).

### Software

**"POEM"**A Python based software for the computation of SONC and SOS certificates in polynomial optimization.

with H. Seidler.

Latest Version: (0.2.1.0(a)). Prototype; for test purposes only. Latest Update: 07/04/19. Development: July 2017 - present.**"Maximal_Mediated_Sets.sage "**.

A SAGE class for the computation of maximal mediated sets.

with J. Hartzer.

Latest Version: (0.1.0.1(a)). Latest Update: 04/03/17. Development: Fall 2016 - present.**"Lopsided_Amoeba_Approximation"**.

A SINGULAR/SAGE software package for the approximation of amoebas via lopsided amoebas and cyclic resultants.

Latest Version: (0.1.0.0(a)). Latest Update: 08/30/16. Development: Spring 2016 - present.**"Viro.sage"**.

A SAGE class for working with Viro's Patchworking

with C. O'Neill and E. Owusu Kwaakwah (alumna).

Latest Version: (0.4b). Latest Update: 05/09/18. Development: Spring 2015 - present.