numericalNoetherianOperators(I)
The method computes specialized Noetherian operators from many sampled points, and attempts to find fitting rational functions using rational function interpolation.
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The behavior of the function can be adjusted using options. Currently only the option DependentSet is required. The following are supported:
TrustedPoint => a point. The function does not compute specialized Noetherian operators from scratch for each point. Instead, it computes it for a "trusted" point on the variety, and uses the obtained Noetherian operators as a template for the rest of the computation. If TrustedPoint is unset, the first point returned by the sampler will be used as the trusted point.
NoetherianDegreeLimit => a non-negative integer. Limits the degrees of the Noetherian operators (with respect to the variables $dx_i$). If unset, will compute the full Noetherian operators of the "trusted" point. Can introduce speed-ups when the maximal degree of the Noetherian operators is known in advance.
Tolerance => a positive real number. This specifies the numerical precision when computing the specialized Noetherian operators. The default value is 1e-6. See Tolerance (NoetherianOperators).
Sampler => a function, taking inputs (n,I), where I is an ideal, and n is an integer. The sampler function returns a list of n points on the component of interest of I. If unset, the default sampler uses Bertini, and assumes that I is primary.
DependentSet => a list of variables that are algebraically dependent. See DependentSet for details.
InterpolationTolerance => a positive real number. This specifies the numerical precision for the interpolation routines. The default value is 1e-6. See Tolerance (NoetherianOperators).
InterpolationDegreeLimit => a non-negative integer. Limits the degree of the interpolated rational function coefficients. If no rational functions are found within the degree limit, outputs an incomplete differential operator
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The object numericalNoetherianOperators is a method function with options.