scholarly journals Numerical algebraic geometry and algebraic kinematics

Acta Numerica ◽  
2011 ◽  
Vol 20 ◽  
pp. 469-567 ◽  
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.

scholarly journals Multiprojective witness sets and a trace test

2020 ◽  
Vol 20 (3) ◽  
pp. 297-318 ◽  
Author(s):  
Jonathan D. Hauenstein ◽  
Jose Israel Rodriguez
Keyword(s):  
Algebraic Geometry ◽  
Polynomial Equations ◽  
Numerical Algebraic Geometry ◽  
Dimensional Solution ◽  
New Approach ◽  
Solution Sets ◽  
Irreducible Decomposition ◽  
Case Examples ◽  
Trace Test ◽  
Systems Of Polynomial Equations

AbstractIn the field of numerical algebraic geometry, positive-dimensional solution sets of systems of polynomial equations are described by witness sets. In this paper, we define multiprojective witness sets which encode the multidegree information of an irreducible multiprojective variety. Our main results generalise the regeneration solving procedure, a trace test, and numerical irreducible decomposition to the multiprojective case. Examples are included to demonstrate this new approach.

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 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].

Complete Solution to the Eight-Point Path Generation of Slider-Crank Four-Bar Linkages

2010 ◽  
Vol 132 (8) ◽  
Author(s):  
Hafez Tari ◽  
Hai-Jun Su
Keyword(s):  
Complete Solution ◽  
Solution Process ◽  
Polynomial Equations ◽  
Homotopy Methods ◽  
Degree Polynomial ◽  
Fourth Degree ◽  
Augmented System ◽  
Polynomial Curve ◽  
Four Bar Linkage ◽  
Classical Homotopy

We study the synthesis of a slider-crank four-bar linkage whose coupler point traces a set of predefined task points. We report that there are at most 558 slider-crank four-bars in cognate pairs passing through any eight specified task points. The problem is formulated for up to eight precision points in polynomial equations. Classical elimination methods are used to reduce the formulation to a system of seven sixth-degree polynomials. A constrained homotopy technique is employed to eliminate degenerate solutions, mapping them to solutions at infinity of the augmented system, which avoids tedious post-processing. To obtain solutions to the augmented system, we propose a process based on the classical homotopy and secant homotopy methods. Two numerical examples are provided to verify the formulation and solution process. In the second example, we obtain six slider-crank linkages without a branch or an order defect, a result partially attributed to choosing design points on a fourth-degree polynomial curve.

scholarly journals Theta Surfaces

2020 ◽  
Author(s):  
Daniele Agostini ◽  
Türkü Özlüm Çelik ◽  
Julia Struwe ◽  
Bernd Sturmfels
Keyword(s):  
Algebraic Geometry ◽  
Theta Functions ◽  
Minkowski Sum ◽  
Zero Set ◽  
Analytic Surface ◽  
Numerical Algebraic Geometry ◽  
Abelian Integrals ◽  
Space Curves ◽  
Quartic Curves ◽  
Special Plane

Abstract A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that any analytic surface that is the Minkowski sum of two space curves in two different ways is a theta surface. The four space curves that generate such a double translation structure are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions.

Analysis of secret sharing schemes based on Nielsen transformations

2018 ◽  
Vol 0 (0) ◽  
Author(s):  
Matvei Kotov ◽  
Dmitry Panteleev ◽  
Alexander Ushakov
Keyword(s):  
Computer Algebra ◽  
Secret Sharing ◽  
Computer Algebra System ◽  
Cryptographic Protocols ◽  
Secret Sharing Schemes ◽  
Polynomial Equations ◽  
Polynomial Size ◽  
Security Properties ◽  
System Of Polynomial Equations

Abstract We investigate security properties of two secret-sharing protocols proposed by Fine, Moldenhauer, and Rosenberger in Sections 4 and 5 of [B. Fine, A. Moldenhauer and G. Rosenberger, Cryptographic protocols based on Nielsen transformations, J. Comput. Comm. 4 2016, 63–107] (Protocols I and II resp.). For both protocols, we consider a one missing share challenge. We show that Protocol I can be reduced to a system of polynomial equations and (for most randomly generated instances) solved by the computer algebra system Singular. Protocol II is approached using the technique of Stallings’ graphs. We show that knowledge of {m-1} shares reduces the space of possible values of a secret to a set of polynomial size.

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