Approximate Polynomial GCD by Approximate Syzygies

One way to compute a GCD of a pair of multivariate polynomials is by finding a certain syzygy. We can weaken this to create an “approximate syzygy”, for the purpose of computing an approximate GCD. The primary tools are Gröbner bases and optimization. Depending on specifics of the formulation, one m...

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Bibliographic Details
Published in:Mathematics in computer science Vol. 13; no. 4; pp. 517 - 532
Main Author: Lichtblau, Daniel
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.12.2019
Springer Nature B.V
Subjects:
Computer Science
Linear programming
Mathematics
Mathematics and Statistics
Optimization
Polynomials
Quadratic programming
Quotients
Hybrid symbolic-numeric algorithms
68W30
Numeric Gröbner basis
Approximate polynomial GCD
13P10
Syzygies
Linear programming
13P05
68W25
Quadratic programming
ISSN:1661-8270, 1661-8289
Online Access:Get full text
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Summary:One way to compute a GCD of a pair of multivariate polynomials is by finding a certain syzygy. We can weaken this to create an “approximate syzygy”, for the purpose of computing an approximate GCD. The primary tools are Gröbner bases and optimization. Depending on specifics of the formulation, one might use quadratic programming, linear programming, unconstrained with quadratic main term and quartic penalty, or a penalty-free sum-of-squares optimization. There are relative strengths and weaknesses to all four approaches, trade-offs in terms of speed vs. quality of result, size of problem that can be handled, and the like. Once a syzygy is found, there is a polynomial quotient to form, in order to get an approximation to an exact quotient. This step too can be tricky and requires careful handling. We will show what seem to be reasonable formulations for the optimization and quotient steps. We illustrate with several examples from the literature.
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ISSN:1661-8270
1661-8289
DOI:10.1007/s11786-019-00392-w