Recognizing Graph Theoretic Properties with Polynomial Ideals

Jesús A. De Loera; Christopher J. Hillar; Peter N. Malkin; Mohamed Omar

doi:10.37236/386

Recognizing Graph Theoretic Properties with Polynomial Ideals

The Electronic Journal of Combinatorics ◽

10.37236/386 ◽

2010 ◽

Vol 17 (1) ◽

Cited By ~ 4

Author(s):

Jesús A. De Loera ◽

Christopher J. Hillar ◽

Peter N. Malkin ◽

Mohamed Omar

Keyword(s):

Algebraic Geometry ◽

Convex Programming ◽

Linear Algebra ◽

Finite Fields ◽

Combinatorial Problems ◽

Polynomial Method ◽

Polynomial Equations ◽

Graph Theoretic ◽

Polynomial Ideals ◽

System Of Polynomial Equations

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect $k$-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.

Download Full-text

Numerical algebraic geometry and algebraic kinematics

Acta Numerica ◽

10.1017/s0962492911000067 ◽

2011 ◽

Vol 20 ◽

pp. 469-567 ◽

Cited By ~ 34

Author(s):

Charles W. Wampler ◽

Andrew J. Sommese

Keyword(s):

Algebraic Geometry ◽

Polynomial Equations ◽

Numerical Algebraic Geometry ◽

Homotopy Methods ◽

Algebraic Framework ◽

Kinematic Studies ◽

Testing Algorithms ◽

System Of Polynomial Equations

In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical algebraic geometry applies broadly to any system of polynomial equations, algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work.

Download Full-text

An Algebraic Exploration of Dominating Sets and Vizing's Conjecture

The Electronic Journal of Combinatorics ◽

10.37236/2194 ◽

2012 ◽

Vol 19 (2) ◽

Cited By ~ 4

Author(s):

Susan Margulies ◽

I. V. Hicks

Keyword(s):

Dominating Set ◽

Hamiltonian Path ◽

Independent Set ◽

Combinatorial Problems ◽

Polynomial Equations ◽

Dominating Sets ◽

Critical Graphs ◽

System Of Polynomial Equations ◽

High Degree ◽

Systems Of Polynomial Equations

Systems of polynomial equations are commonly used to model combinatorial problems such as independent set, graph coloring, Hamiltonian path, and others. We formulate the dominating set problem as a system of polynomial equations in two different ways: first, as a single, high-degree polynomial, and second as a collection of polynomials based on the complements of domination-critical graphs. We then provide a sufficient criterion for demonstrating that a particular ideal representation is already the universal Grobner bases of an ideal, and show that the second representation of the dominating set ideal in terms of domination-critical graphs is the universal Grobner basis for that ideal. We also present the first algebraic formulation of Vizing's conjecture, and discuss the theoretical and computational ramifications to this conjecture when using either of the two dominating set representations described above.

Download Full-text

Low Degree Nullstellensatz Certificates for 3-Colorability

The Electronic Journal of Combinatorics ◽

10.37236/5103 ◽

2016 ◽

Vol 23 (1) ◽

Author(s):

Bo Li ◽

Benjamin Lowenstein ◽

Mohamed Omar

Keyword(s):

Linear Algebra ◽

Directed Graphs ◽

Polynomial Equations ◽

Combinatorial Characterization ◽

Seminal Paper ◽

Low Degree ◽

System Of Polynomial Equations

In a seminal paper, De Loera et. al introduce the algorithm NulLA (Nullstellensatz Linear Algebra) and use it to measure the difficulty of determining if a graph is not 3-colorable. The crux of this relies on a correspondence between 3-colorings of a graph and solutions to a certain system of polynomial equations over a field $\mathbb{k}$. In this article, we give a new direct combinatorial characterization of graphs that can be determined to be non-3-colorable in the first iteration of this algorithm when $\mathbb{k}=GF(2)$. This greatly simplifies the work of De Loera et. al, as we express the combinatorial characterization directly in terms of the graphs themselves without introducing superfluous directed graphs. Furthermore, for all graphs on at most $12$ vertices, we determine at which iteration NulLA detects a graph is not 3-colorable when $\mathbb{k}=GF(2)$.

Download Full-text

Algebraically closed commutative rings

Journal of Symbolic Logic ◽

10.2307/2273048 ◽

1973 ◽

Vol 38 (3) ◽

pp. 493-499 ◽

Cited By ~ 9

Author(s):

G. L. Cherlin

Keyword(s):

Commutative Ring ◽

Semiprime Ring ◽

Commutative Rings ◽

Finite System ◽

Polynomial Equations ◽

Finitely Generated ◽

Model Companion ◽

Polynomial Ideals ◽

System Of Polynomial Equations ◽

Hilbert Nullstellensatz

A commutative ring A is said to be algebraically closed if every finite system of polynomial equations and inequations in one or more variables with coefficients in A which has a solution in some (commutative) extension of A already has a solution in A. Abraham Robinson's study of model-theoretic forcing has provided powerful new tools for the study of algebraically closed structures in general, and will be applied here to the study of algebraically closed commutative rings. Familiarity with the model-theoretic notions connected with the study of algebraically closed structures is assumed; for background consult [1], [2], and [3].Our main results are the following:1. The theory of commutative rings with identity has no model companion in the sense of Robinson.2. The Hilbert Nullstellensatz, suitably formulated for the class of algebraically closed commutative rings, holds for finitely generated polynomial ideals but fails for certain infinitely generated polynomial ideals.3. If A is algebraically closed, then A/rad A need not be algebraically closed as a semiprime ring: If A is finitely generic then A/rad A is algebraically closed as a semiprime ring, but if A is infinitely generic then A/rad A is not algebraically closed as a semiprime ring.

Download Full-text

Public-key cryptosystem based on invariants of diagonalizable groups

Groups – Complexity – Cryptology ◽

10.1515/gcc-2017-0003 ◽

2017 ◽

Vol 9 (1) ◽

Author(s):

František Marko ◽

Alexandr N. Zubkov ◽

Martin Juráš

Keyword(s):

Linear Algebra ◽

Finite Fields ◽

Number Fields ◽

Euclidean Algorithm ◽

Public Key ◽

Public Key Cryptosystem ◽

Finite Rings

AbstractWe develop a public-key cryptosystem based on invariants of diagonalizable groups and investigate properties of such a cryptosystem first over finite fields, then over number fields and finally over finite rings. We consider the security of these cryptosystem and show that it is necessary to restrict the set of parameters of the system to prevent various attacks (including linear algebra attacks and attacks based on the Euclidean algorithm).

Download Full-text

A Numerical Local Dimension Test for Points on the Solution Set of a System of Polynomial Equations

SIAM Journal on Numerical Analysis ◽

10.1137/08073264x ◽

2009 ◽

Vol 47 (5) ◽

pp. 3608-3623 ◽

Cited By ~ 28

Author(s):

Daniel J. Bates ◽

Jonathan D. Hauenstein ◽

Chris Peterson ◽

Andrew J. Sommese

Keyword(s):

Solution Set ◽

Polynomial Equations ◽

Local Dimension ◽

System Of Polynomial Equations

Download Full-text

Bivariate systems of polynomial equations with roots of high multiplicity

Journal of Algebra and Its Applications ◽

10.1142/s0219498823500147 ◽

2021 ◽

Author(s):

I. Nikitin

Keyword(s):

High Multiplicity ◽

Polynomial Equations ◽

Support Sets ◽

Multiplicity Formula ◽

System Of Polynomial Equations ◽

Systems Of Polynomial Equations

Given a bivariate system of polynomial equations with fixed support sets [Formula: see text] it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity [Formula: see text] for all [Formula: see text] in the range [Formula: see text], where [Formula: see text] is the set of all integral vectors that shift B to a subset of [Formula: see text]. As an application, we classify all pairs [Formula: see text] such that the system supported at [Formula: see text] does not have a solution of multiplicity higher than [Formula: see text].

Download Full-text