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 Grobner 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.