Computer Algebra in Scientific Computing. 25th International Workshop, CASC 2023 Havana, Cuba, August 28 – September 1, 2023 Proceedings

Preface
Acknowledgments
Organization
Abstracts of Invited Talks
	Normal Forms of Integer Matrices
	On the Performance of Local Search Algorithms for K-SAT Problems in Random Graphs
	Contents
Computing GCDs of Multivariate Polynomials over Algebraic Number Fields Presented with Multiple Extensions
	1 Introduction
		1.1 Motivation for the Algorithm
		1.2 Preliminaries
		1.3 Paper Outline
	2 Converting Q(1,…,n) to a Single Extension Q()
		2.1 Computing a Primitive Element and its Minimal Polynomial
		2.2 The Isomorphism
	3 The Modular Gcd Algorithm
		3.1 PGCD
		3.2 MGCD
	4 Complexity
	5 Implementation
		5.1 Maple Implementation
		5.2 Benchmark
	6 Conclusion and Future Work
	References
Generating Elementary Integrable Expressions
	1 Introduction
		1.1 Machine Learning and Computer Algebra
		1.2 Symbolic Integration Meta-Algorithms
		1.3 Motivation
		1.4 Contributions and Plan
	2 Existing Datasets and Data Generation Methods
		2.1 Deep Learning for Symbolic Mathematics
		2.2 Other Existing Datasets
	3 The Risch Algorithm
		3.1 The Rational Part
		3.2 The Polynomial Part
	4 Data Generation Based on the Risch Algorithm
		4.1 Polynomial Integrable Expressions
		4.2 Rational Integrable Expressions
	5 Discussion
		5.1 Risch Data Generation Benefits
		5.2 Future Work
	References
How to Automatise Proofs of Operator Statements: Moore–Penrose Inverse; A Case Study
	1 Introduction
	2 From Operator Identities to Noncommutative Polynomials
	3 Treating Existential Statements
	4 Treating Common Properties
		4.1 Real Matrices
		4.2 Identity Operators
		4.3 Injectivity, Surjectivity, and Full Matrix Ranks
		4.4 Range Inclusions
	5 Logical Framework
	6 Case Study
	A The Software Package Operator_gb
		A.1 Certifying Operator Statements
		A.2 Useful Auxiliary Functions for Treating Operator Statements
		A.3 Quivers and Detecting Typos
		A.4 Gröbner Basis Computations
		A.5 Heuristics for Finding Polynomials of Certain Form
	References
A Modular Algorithm for Computing the Intersection of a One-Dimensional Quasi-Component and a Hypersurface
	1 Introduction
	2 Preliminaries
	3 Genericity Assumptions
	4 The Modular Method
		4.1 The Fumber of Bad Specializations is Finite
		4.2 Number of Bad Specializations and Other Degree Estimates
		4.3 A Modular Algorithm
	5 Relaxing the Hypotheses
	6 Implementation
	7 Experimentation and Discussion
	References
Certified Study of Internal Solitary Waves
	1 Introduction
	2 Improved Serre-Like Model
	3 Steady Motions
	4 Algebraic Analysis and Symbolic Computations
		4.1 Improved SGN
		4.2 Expressions Related to D, with 1 and 2
		4.3 Illustrative Case
		4.4 Classical SGN, i.e., 1 =2=0
	5 Partition of the Parameters Space
	6 Phase Plane Analysis
		6.1 Local Analysis
		6.2 Global Analysis
	7 An Explicit Example of Slug
	8 Conclusion
	References
Root-Squaring for Root-Finding
	1 Introduction
		1.1 Polynomial Root-Finding
		1.2 Classical Root-Squaring Iterations
		1.3 Related Works on Root-Squaring and Its Applications
		1.4 The Two Nearly Optimal Polynomial Root-Finders
		1.5 Our Contribution
		1.6 Organization of Our Paper
	2 Background and Motivation
		2.1 Definitions
		2.2 Extension of the DLG Iterations
		2.3 NIRp, Root-Squaring, and Estimation of Extremal Root Radii
		2.4 NIRp, Root-Squaring, and Recent E/i Tests
		2.5 Recovery of Complex Roots
	3 Our Root-Squaring Algorithm
		3.1 Implementation Details
		3.2 Analysis
	4 Experimental Results
		4.1 Setup
		4.2 Our Findings
		4.3 Alternative Bounds on Extremal Root Radii
	5 Conclusion
	A Additional Tables
	References
Symbolic-Numerical Algorithm for Solving the Problem of Heavy Ion Collisions in an Optical Model with a Complex Potential
	1 Introduction
	2 Optical Model and IWBC Model in the Single-Channel Approximation
	3 The Optical Model Algorithm
	4 Benchmark Calculations
	5 Conclusions
	References
On the Complexity of Linear Algebra Operations over Algebraic Extension Fields
	1 Introduction
	2 Preliminaries
	3 Complexity Results
		3.1 Multiplication Table
		3.2 Algebraic Inverse
		3.3 Gaussian Elimination
		3.4 Minimal Polynomial
	4 Notes on Implementation and Experimental Results
		4.1 Dependence on Matrix Dimension
		4.2 Dependence on Normal Set Size
	References
Range Functions of Any Convergence Order and Their Amortized Complexity Analysis
	1 Introduction
		1.1 Why We Must Extend the CL Framework
		1.2 Overview
		1.3 Terminology and Notation
	2 Generalized CL Framework
		2.1 Achieving Any Order of Convergence
		2.2 Strong Box Functions
	3 A Practical Range Function of Order 4
	4 Holistic Complexity Analysis of Range Functions
		4.1 Amortized Complexity of
		4.2 Amortized Complexity of
		4.3 Amortized Complexity for Hermite Schemes
	5 Experimental Results
		5.1 Non-maximal Recursion Levels
	6 Conclusions and Future Work
	References
Stability and Zero-Hopf Bifurcation Analysis of the Lorenz–Stenflo System Using Symbolic Methods
	1 Introduction and Main Results
	2 Preliminary Results
	3 Stability Conditions of the Lorenz–Stenflo System
	4 Zero-Hopf Bifurcation of the Lorenz–Stenflo System
	5 Zero-Hopf Bifurcation in a Special Lorenz–Stenflo System
	6 Conclusions
	References
Non-principal Branches of Lambert W. A Tale of 2 Circles
	1 Introduction
		1.1 Definitions
		1.2 Expansions
		1.3 Branch Structure
		1.4 Asymptotic Expansions
		1.5 Outline
	2 de Bruijn Series for Large z
	3 de Bruijn Series for Small z
	4 A Surprising Convergence
	5 A Further Variation
	6 Concluding Remarks
	References
On the Qualitative Analysis of the Equations of Motion of a Nonholonomic Mechanical System
	1 Introduction
	2 Problem Statement
	3 On Stationary Sets in the Case of Absence of External Forces
	4 On Stationary Sets in the Case of the Presence of External Forces
	5 On Pendulum-Like Motions
	6 On the Stability of Stationary Sets
		6.1 The Case of Absence of External Forces
		6.2 The Case of the Presence of External Forces
	7 Conclusion
	References
Solving Parametric Linear Systems Using Sparse Rational Function Interpolation
	1 Introduction
	2 Sparse Multivariate Rational Function Interpolation
		2.1 Cuyt and Lee\'s Algorithm
		2.2 Using a Kronecker Substitution on the Parameters
	3 The Algorithm
	4 Analysis
		4.1 Failure Probability Analysis
		4.2 Complexity Analysis
	5 Implementation and Benchmarks
	References
On the Distance to the Nearest Defective Matrix
	1 Introduction
	2 Algebraic Preliminaries
	3 Complex Matrix
	4 Real Matrix
	5 Conclusions
	References
Effective Algorithm for Computing Noetherian Operators of Positive Dimensional Ideals
	1 Introduction
	2 Noetherian Operators of Zero Dimensional Ideals
	3 Mathematical Basics
		3.1 Extension and Contraction
		3.2 Noetherian Operators of a Primary Ideal qe K(U)[Y]
	4 Main Results
		4.1 Generalization
		4.2 Comparisons
		4.3 Computing Noetherian Representations
	References
On the Structure and Generators of Differential Invariant Algebras
	1 Introduction
	2 Multi-indices
	3 The Jet Calculus
	4 Invariantization
	5 The Recurrence Formulae
	6 The Symbolic Invariant Calculus
	7 The Extended Symbolic Invariant Calculus
	8 Independence
	9 Generating Differential Invariants
	10 The Algorithm
	References
An Algorithm for the Intersection Problem of Planar Parametric Curves
	1 Introduction
	2 Reduction Strategy
		2.1 Reduction Strategy
		2.2 Preconditioner
	3 Uniqueness and Existence
		3.1 An Opposite Monotone System in a Box
		3.2 How to Transform a System to an Opposite Monotone System in a Box
		3.3 How to Check the Existence
	4 Some Singular Cases
		4.1 Cusp Cases
		4.2 Self-intersection Cases
		4.3 Tangential Cases
		4.4 Mixed Cases
	5 Algorithm
	6 Experiments
	7 Conclusion
	References
A Symbolic-Numeric Method for Solving the Poisson Equation in Polar Coordinates
	1 Introduction
	2 The CLS Method for the Numerical Solution of the Poisson Equation in Polar Coordinates
	3 Computational Results
	4 Conclusions
	References
Two Variants of Bézout Subresultants for Several Univariate Polynomials
	1 Introduction
	2 Preliminaries
		2.1 The Bézout-Type Subresultant and Its Variants for Two Polynomials
		2.2 Subresultant in Roots for Several Polynomials
	3 Main Results
	4 Proof
		4.1 Proof of Theorem 2-(1)
		4.2 Proof of Theorem 2-(2)
	5 Experimental Results
	6 Conclusion
	References
Efficient Quotients of Non-commutative Polynomials
	1 Introduction
	2 Background
		2.1 Notation
		2.2 Division
		2.3 Whole Shift and Whole Shifted Inverse
	3 Division in Non-Commutative R[x]
		3.1 Definitions and Classical Algorithms
		3.2 Whole Shift and Whole Shifted Inverse in R[x]
		3.3 Quotients from the Whole Shifted Inverse in R[x]
	4 Generic Algorithm for the Whole Shifted Inverse
	5 Non-commutative Polynomial Example
	6 Division in R[x; , ]
		6.1 Definitions and Classical Algorithms
		6.2 Whole Shift and Inverse in R[x; , ]
		6.3 Quotients from Whole Shifted Inverses in R[x; , ]
	7 Skew Polynomial Examples
		7.1 Differential Operators
		7.2 Difference Operators
		7.3 Difference Operators with Matrix Coefficients
	8 Conclusions
	References
Inverse Kinematics and Path Planning of Manipulator Using Real Quantifier Elimination Based on Comprehensive Gröbner Systems
	1 Introduction
	2 Inverse Kinematics of a 3-DOF Robot Manipulator
	3 Real Quantifier Elimination Based on CGS
		3.1 CGS
		3.2 Real Root Counting
		3.3 CGS-QE Algorithm
	4 Solving the Inverse Kinematic Problem
		4.1 Removing a Segment Not Existing in R3
		4.2 Calculating the Number of Real Roots
		4.3 Calculation for Non-Zero Dimensional Ideals
		4.4 Experiments
	5 Path and Trajectory Planning
		5.1 Path and Trajectory Planning for a Path Expressed as a Function of Time
		5.2 Trajectory Planning with Verification of the Feasibility of the Inverse Kinematic Solution
	6 Concluding Remarks
	References
Author Index