Related papers: Combinatorial Resultants in the Algebraic Rigidity…
We present an algorithm for computing circuit polynomials in the algebraic rigidity matroid $\mathcal{A}(\text{CM}_n)$ associated to the Cayley-Menger ideal CM$_n$ for $n$ points in 2D. It relies on combinatorial resultants, a new operation…
A classical problem in Distance Geometry, with multiple practical applications (in molecular structure determination, sensor network localization etc.) is to find the possible placements of the vertices of a graph with given edge lengths.…
A rigidity circuit (in 2D) is a minimal dependent set in the rigidity matroid, i.e. a minimal graph supporting a non-trivial stress in any generic placement of its vertices in $\mathbb R^2$. Any rigidity circuit on $n\geq 5$ vertices can be…
Combinatorial rigidity theory seeks to describe the rigidity or flexibility of bar-joint frameworks in R^d in terms of the structure of the underlying graph G. The goal of this article is to broaden the foundations of combinatorial rigidity…
This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a)…
This is a continuation of the series of notes on the dynamics of quadratic polynomials. We show the following Rigidity Theorem: Any combinatorial class contains at most one quadratic polynomial satisfying the secondary limbs condition with…
This paper investigates the independence polynomials arising from iterated strong products of cycle graphs, examining their algebraic symmetries and combinatorial structures. Leveraging modular arithmetic and Galois theory, we establish…
We establish a connection between problems studied in rigidity theory and matroids arising from linear algebraic constructions like tensor products and symmetric products. A special case of this correspondence identifies the problem of…
The representation of polynomials by arithmetic circuits evaluating them is an alternative data structure which allowed considerable progress in polynomial equation solving in the last fifteen years. We present a circuit based computation…
An affine variety induces the structure of an algebraic matroid on the set of coordinates of the ambient space. The matroid has two natural decorations: a circuit polynomial attached to each circuit, and the degree of the projection map to…
Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry…
We study the rigidity of body-and-cad frameworks which capture the majority of the geometric constraints used in 3D mechanical engineering CAD software. We present a combinatorial characterization of the generic minimal rigidity of a subset…
We present a novel algebraic combinatorial view on low-rank matrix completion based on studying relations between a few entries with tools from algebraic geometry and matroid theory. The intrinsic locality of the approach allows for the…
In this paper an algorithm is given to determine all possible structurally different linearly conjugate realizations of a given kinetic polynomial system. The solution is based on the iterative search for constrained dense realizations…
The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic…
We describe a combinatorial approach for investigating properties of rational numbers. The overall approach rests on structural bijections between rational numbers and familiar combinatorial objects, namely rooted trees. We emphasize that…
This dissertation presents new results on three different themes all related to matroid polytopes. First we investigate properties of Ehrhart polynomials of matroid polytopes, independence matroid polytopes, and polymatroids. We prove that…
We present a novel preconditioning technique for proximal optimization methods that relies on graph algorithms to construct effective preconditioners. Such combinatorial preconditioners arise from partitioning the graph into forests. We…
We develop certain combinatorial tools for the study of discriminants of general systems of polynomial equations. Applying these tools in a sequel paper, we completely classify components of such discriminants, generalizing the classical…
We consider efficient route planning for robots in applications such as infrastructure inspection and automated surgical imaging. These tasks can be modeled via the combinatorial problem Graph Inspection. The best known algorithms for this…