scholarly journals Low Degree Nullstellensatz Certificates for 3-Colorability

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 $\mathbb{k}=GF(2)$.

scholarly journals Solving equations over small unary algebras

2005 ◽  
Vol DMTCS Proceedings vol. AF,... (Proceedings) ◽  
Author(s):  
Przemyslaw Broniek
Keyword(s):  
Polynomial Time ◽  
Partial Characterization ◽  
Polynomial Equations ◽  
Solving Equations ◽  
International Audience ◽  
Np Complete ◽  
System Of Polynomial Equations

International audience We consider the problem of solving a system of polynomial equations over fixed algebra $A$ which we call MPolSat($A$). We restrict ourselves to unary algebras and give a partial characterization of complexity of MPolSat($A$). We isolate a preorder $P(A)$ to show that when $A$ has at most 3 elements then MPolSat($A$) is in $P$ when width of $P(A)$ is at most 2 and is NP-complete otherwise. We show also that if $P ≠ NP$ then the class of unary algebras solvable in polynomial time is not closed under homomorphic images.

scholarly journals Recognizing Graph Theoretic Properties with Polynomial Ideals

10.37236/386 ◽  
2010 ◽  
Vol 17 (1) ◽  
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.

scholarly journals A fault attack on the Niederreiter cryptosystem using binary irreducible Goppa codes

2020 ◽  
Vol Volume 12, issue 1 ◽  
Author(s):  
Julian Danner ◽  
Martin Kreuzer
Keyword(s):  
Fault Injection ◽  
Polynomial Equations ◽  
Secret Key ◽  
Goppa Codes ◽  
Fault Attack ◽  
Low Degree ◽  
Secret Keys ◽  
Niederreiter Cryptosystem ◽  
System Of Polynomial Equations ◽  
Solving Strategy

A fault injection framework for the decryption algorithm of the Niederreiter public-key cryptosystem using binary irreducible Goppa codes and classical decoding techniques is described. In particular, we obtain low-degree polynomial equations in parts of the secret key. For the resulting system of polynomial equations, we present an efficient solving strategy and show how to extend certain solutions to alternative secret keys. We also provide estimates for the expected number of required fault injections, apply the framework to state-of-the-art security levels, and propose countermeasures against this type of fault attack. Comment: 20 pages

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

2009 ◽  
Vol 47 (5) ◽  
pp. 3608-3623 ◽  
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

scholarly journals Bivariate systems of polynomial equations with roots of high multiplicity

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].

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