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Geometric modeling by constraints leads to large systems of algebraic equations. This paper studies bipartite graphs underlaid by systems of equations. It shows how these graphs make possible to polynomially decompose these systems into…
A Hamilton decomposition of a graph is a partitioning of its edge set into disjoint spanning cycles. The existence of such decompositions is known for all hypercubes of even dimension $2n$. We give a decomposition for the case $n = 2^a3^b$…
This paper presents a method for constructing flat deformations of associative algebras. We will refer to this method as method two because it is a generalisation of the method obtained in [1]. The deformations obtained using the first two…
Reed showed that, if two graphs are $P_4$-isomorphic, then either both are perfect or none of them is. In this note we will derive an analogous result for perfect digraphs.
Representations of nonnegative polynomials as sums of squares are central to real algebraic geometry and the subject of active research. The sum-of-squares representations of a given polynomial are parametrized by the convex body of…
A decomposition of a graph is a set of subgraphs whose edges partition those of $G$. The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011 states that every connected cubic graph can be decomposed into a spanning tree, a…
A graph $G = (V, E)$ is word-representable, if there exists a word w over the alphabet V such that for letters ${x, y} \in V$ , $x$ and $y$ alternate in $w$ if and only if $xy \in E$. In this paper, we prove that any non-empty…
We give an introduction to the study of algebraic hypersurfaces, focusing on the problem of when two hypersurfaces are isomorphic or close to being isomorphic. Working with hypersurfaces and emphasizing examples makes it possible to discuss…
We study a numerical semigroup ring as an algebra over another numerical semigroup ring. The complete intersection property of numerical semigroup algebras is investigated using factorizations of monomials into minimal ones. The goal is to…
Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue:…
An open question in the theory of word-representable graphs for the past decade has been whether the line graph of a non-word-representable graph is always non-word-representable. By formulating an appropriate optimization problem for the…
This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization.
We investigate novel random graph embeddings that can be computed in expected polynomial time and that are able to distinguish all non-isomorphic graphs in expectation. Previous graph embeddings have limited expressiveness and either cannot…
We study isometric representations of product systems of correspondences over the semigroup $\mathbb{N}^k$ which are minimal dilations of finite dimensional, fully coisometric representations. We show the existence of a unique minimal…
How to draw the vertices of a complete multipartite graph $G$ on different points of a bounded $d$-dimensional integer grid, such that the sum of squared distances between vertices of $G$ is (i) minimized or (ii) maximized? For both…
Graph-based semantic representations are valuable in natural language processing, where it is often simple and effective to represent linguistic concepts as nodes, and relations as edges between them. Several attempts has been made to find…
A realization of a graph $G=(V,E)$ is a map $v\colon V\to\Bbb R^d$ that assigns to each vertex a point in $d$-dimensional Euclidean space. We study graph realizations from the perspective of representation theory (expressing certain…
A graphical expansion formula for non-commutative matrix integrals with values in a finite-dimensional real or complex von Neumann algebra is obtained in terms of ribbon graphs and their non-orientable counterpart called Moebius graphs. The…
We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices. These graphs have an interface made of…
We study cut algebras which are toric rings associated to graphs. The key idea is to consider suitable retracts to understand algebraic properties and invariants of such algebras like being a complete intersection, having a linear…