Approximate polynomial GCD by approximate syzygies, redux (Dr. Daniel Lichtblau)
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I will show in brief how one can compute the GCD of a pair of multivariate polynomials by finding a syzygy. I will then show how we can weaken this and create an "approximate syzygy" to find an approximate GCD. The primary tools are Gröbner bases and some flavor of 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, and it is 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.