5.1 Elimination theory and its applications

Elimination theory is a powerful tool in algebraic geometry for simplifying polynomial systems. It helps us find conditions for solvability and reduce the number of variables, making complex problems more manageable. This is crucial for solving equations and studying algebraic varieties.

Resultants and Gröbner bases are key techniques in elimination theory. They allow us to eliminate variables and project algebraic varieties onto lower-dimensional spaces. These methods have wide-ranging applications in computer algebra, geometric modeling, robotics, and computer vision.

Elimination theory fundamentals

Key concepts and goals

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• Elimination theory is a branch of algebraic geometry that deals with the problem of eliminating variables from a system of polynomial equations to obtain a reduced system that is easier to solve
• The main goal of elimination theory is to find necessary and sufficient conditions for the solvability of a system of polynomial equations, expressed in terms of the coefficients of the equations
• Elimination theory plays a crucial role in computational algebraic geometry, as it provides algorithmic methods for solving polynomial systems and studying their solution sets (algebraic varieties)
• The fundamental idea behind elimination is to derive new equations that do not contain certain variables, effectively reducing the dimensionality of the problem

Historical context and complexity

• Elimination theory has its roots in the works of mathematicians such as Bézout, Sylvester, and Cayley, who developed early methods for eliminating variables from polynomial equations
• The complexity of elimination algorithms is a central concern in elimination theory, as the size of the resulting equations can grow exponentially with the number of variables and the degree of the input polynomials
• Developing efficient elimination algorithms and understanding their complexity is an active area of research in computational algebraic geometry
• The choice of elimination method (resultants, Gröbner bases, etc.) depends on the specific structure of the polynomial system and the desired trade-offs between complexity and output size

Elimination techniques for polynomials

Resultants and their formulations

• Resultants are a fundamental tool in elimination theory, used to eliminate a variable from a pair of polynomial equations in one variable
  • The Sylvester resultant is a classic resultant formulation that expresses the resultant as the determinant of a matrix constructed from the coefficients of the polynomials
  • The Bézout resultant is another formulation that uses the Bézout matrix, which is a smaller matrix compared to the Sylvester matrix
• Resultants have the property that they vanish if and only if the polynomials have a common root, providing a criterion for the existence of solutions
• Resultant formulations can be generalized to multivariate polynomials, leading to the concept of multivariate resultants and their matrix representations
• Resultant-based elimination can be used to compute the projection of an algebraic variety onto a lower-dimensional space

Gröbner bases and elimination ideals

• Gröbner bases, introduced by Bruno Buchberger, provide a powerful framework for eliminating variables from polynomial systems with multiple variables
  • A Gröbner basis is a special generating set of an ideal in a polynomial ring, which allows for the systematic elimination of variables using a multivariate division algorithm
  • The choice of monomial ordering in the Gröbner basis computation affects the elimination process and the structure of the resulting equations
• Elimination ideals are ideals obtained by eliminating certain variables from a given ideal, and they play a key role in studying the projection of algebraic varieties
• Gröbner bases can be computed using algorithms such as Buchberger's algorithm or the F4/F5 algorithms, which are implemented in various computer algebra systems
• The computation of Gröbner bases can be computationally expensive, especially for large polynomial systems, and techniques such as modular methods and signature-based algorithms have been developed to improve efficiency

Specialized elimination methods

• Elimination can also be performed using resultant matrices, which are constructed from the coefficients of the polynomial equations and whose entries are polynomials in the remaining variables
• Specialized elimination methods, such as the Dixon resultant or the sparse resultant, have been developed to handle polynomial systems with specific structures or sparse polynomials more efficiently
  • The Dixon resultant is based on the Dixon matrix, which is constructed from the coefficients of the polynomials and has a smaller size compared to the Sylvester matrix
  • The sparse resultant takes advantage of the sparse structure of polynomials (polynomials with many zero coefficients) to reduce the complexity of elimination
• Toric elimination theory is a branch of elimination theory that deals with polynomial systems defined by toric varieties, which are algebraic varieties with a special combinatorial structure
• Homotopy continuation methods, such as polyhedral homotopy or monodromy, can be used to numerically solve polynomial systems by deforming a simpler system into the target system, avoiding the need for explicit elimination

Geometric interpretation of elimination

Algebraic varieties and their projections

• Algebraic varieties are geometric objects defined as the solution sets of systems of polynomial equations, and they are the central objects of study in algebraic geometry
• Elimination theory provides a bridge between the algebraic description of varieties (polynomial equations) and their geometric properties
• The elimination of variables corresponds to projecting an algebraic variety onto a lower-dimensional space, such as projecting a curve or surface onto a line or plane
• The closure of the projection of a variety is itself an algebraic variety, described by the elimination ideal obtained from the original ideal defining the variety

Dimension, singularities, and degree

• The dimension of an algebraic variety is related to the number of variables that cannot be eliminated from its defining equations, and elimination theory helps in determining the dimension and other geometric invariants of varieties
  • The dimension of a variety can be computed by considering the Krull dimension of its coordinate ring or by analyzing the structure of its defining equations
  • Elimination theory provides tools for studying the dimension of projected varieties and understanding the relationship between the dimensions of a variety and its projections
• Singularities of algebraic varieties, such as self-intersections or cusps, can be studied using elimination theory by analyzing the structure of the eliminated equations and their multiplicities
  • Singular points of a variety are points where the tangent space has a higher dimension than the variety itself, and they can be detected by examining the Jacobian matrix of the defining equations
  • Elimination theory helps in understanding the singularities of projected varieties and their relationship to the singularities of the original variety
• The degree of an algebraic variety, which measures its complexity and intersection properties, can be computed using elimination techniques and resultant formulas
  • The degree of a variety is defined as the number of intersection points with a generic linear subspace of complementary dimension
  • Resultant formulas, such as the Bézout theorem or the multihomogeneous Bézout theorem, provide bounds and exact formulas for the degree of a variety in terms of the degrees of its defining equations

Real algebraic geometry and semi-algebraic sets

• Real algebraic geometry is the study of real solutions to polynomial equations and the properties of semi-algebraic sets, which are subsets of real space defined by polynomial inequalities
• Elimination theory plays a role in real algebraic geometry by providing methods for projecting semi-algebraic sets and studying their topological and geometric properties
• Real elimination methods, such as the cylindrical algebraic decomposition (CAD) algorithm, are used to compute the projection of a semi-algebraic set onto a lower-dimensional space and to study the topology of the projected set
• Quantifier elimination is a key problem in real algebraic geometry, which involves eliminating quantifiers (such as "for all" or "there exists") from a formula involving polynomial inequalities, and elimination techniques are used to solve this problem

Elimination theory applications

Computer algebra systems and symbolic computation

• Elimination theory is a fundamental tool in computer algebra systems (CAS) for symbolically solving polynomial systems and performing algebraic computations
  • CAS such as Mathematica, Maple, and SageMath implement various elimination algorithms, including Gröbner bases and resultant methods, to manipulate polynomial equations efficiently
  • Elimination techniques are used in CAS for simplifying expressions, solving equations, and studying the properties of algebraic varieties
• Symbolic integration and summation algorithms often rely on elimination theory to compute closed-form solutions or to determine the existence of elementary integrals or sums
• Elimination theory is used in the symbolic solution of differential equations, where the unknown functions are represented as polynomials and the differential equations are transformed into algebraic equations

Geometric modeling and computer-aided design

• In geometric modeling, elimination theory is used for implicitization, which is the process of converting a parametric representation of a curve or surface into an implicit equation
  • Implicitization is important for tasks such as intersection computation, boundary evaluation, and ray tracing in computer graphics and CAD (computer-aided design) systems
  • Elimination-based methods, such as the resultant or Gröbner basis approach, are commonly used for implicitization of rational curves and surfaces
• Elimination theory is also used in the computation of the topology and singularities of algebraic curves and surfaces, which is important for robust geometric modeling and visualization
• Parametrization, the inverse problem of implicitization, can also be approached using elimination techniques, such as the Gröbner basis method for finding rational parametrizations of algebraic curves and surfaces

Applications in other fields

• Elimination theory is applied in solving inverse kinematics problems in robotics, where the goal is to determine the joint angles of a robot arm given the desired position and orientation of the end effector
  • The inverse kinematics equations often form a system of polynomial equations, which can be solved using elimination techniques to find the possible configurations of the robot arm
  • Elimination theory helps in understanding the singularities and reachable workspaces of robot manipulators, which are important for motion planning and control
• In computer vision and image processing, elimination theory is used for tasks such as camera calibration, 3D reconstruction, and object recognition
  • The equations describing the geometric constraints in these problems often involve polynomial equations, which can be solved or simplified using elimination methods
  • Elimination theory is used in the estimation of camera parameters, the computation of 3D point correspondences, and the fitting of geometric models to image data
• Elimination theory also finds applications in other areas, such as:
  • Cryptography: for solving polynomial systems arising from cryptographic protocols and analyzing the security of cryptographic schemes
  • Coding theory: for decoding algebraic codes and studying the properties of error-correcting codes based on algebraic curves and surfaces
  • Optimization: for solving polynomial optimization problems and studying the geometry of feasible sets and optimal solutions
  • Computational biology: for studying the algebraic structure of biological systems, such as gene regulatory networks or biochemical reaction networks, and for parameter estimation and model selection in systems biology