symbolicWeightEDDegree -- symbolically compute ED degrees of affine varieties

Usage:

UED = determinantalUnitEDDegree(F)

GED = determinantalGenericEDDegree(F)

GED = symbolicWeightEDDegree(F,U,W)
Inputs:
- F, a list, a system of polynomials (system need not be square)
- U, a list, a (generic) data vector
- W, a list, a (generic) weight vector
Optional inputs:
- ReturnCriticalIdeal => ..., default value false, whether to return the ideal of critical points
Outputs:
- ED, an integer, a generic/unit Euclidean distance degree
- ICP, an ideal, ideal of critical points (if ReturnCriticalIdeal is set)

Description

This method computes Euclidean distance (ED) degrees for the variety defined by the system $F$ via symbolic computation. The ideal of critical points is formed by saturating the defining ideal of the variety with minors of the Jacobian and Augmented Jacobian. The degree of this ideal is the ED degree of the variety. The unit variant of this method computes an ED degree using random (integer) data and unit weights, whereas the generic variant will use random data and random weights.

i1 : R = QQ[x,y];

i2 : F = {x^2 + y^2 - 1};

i3 : (U,W) = ({12, 23}, {15, 331});

i4 : UED = determinantalUnitEDDegree F

o4 = 2

i5 : GED = determinantalGenericEDDegree F

o5 = 4

i6 : ICP = symbolicWeightEDDegree(F, U, W, ReturnCriticalIdeal => true)

                                    2    2
o6 = ideal (316x*y - 7613x + 180y, x  + y  - 1)

o6 : Ideal of R

Ways to use symbolicWeightEDDegree:

symbolicWeightEDDegree(List,List,List)

For the programmer

The object symbolicWeightEDDegree is a method function with options.

The source of this document is in EuclideanDistanceDegree.m2:206:0.

symbolicWeightEDDegree -- symbolically compute ED degrees of affine varieties

Description

Menu

Ways to use symbolicWeightEDDegree:

For the programmer