polarVars -- Computes the list of all polar varieties of a variety

Usage:

polarVars(I)
Inputs:
- I, an ideal, in a polynomial ring $k[x_1,\dots, x_n]$ for a field $k$, defining a closed variety in $k^n$ (or in $\PP^n$ if the ideal is homogeneous).
Optional inputs:
- Algorithm (missing documentation) => ..., default value "saturate",
- Print (missing documentation) => ..., default value false,
Outputs:
- pols, a list of ideals defining polar varieties to $X=V(I)$ relative to a random flag of linear subspaces. The list starts with a dimension 0 polar variety and ends with the $dim(X)$ polar variety, which is $X$, hence the last ideal in the list is I.

Description

For a complex variety $X=V(I)$ this command computes a list of ideals defining polar varieties to $X=V(I)$ relative to a random flag of linear subspaces. For the relevant definitions see, e.g., [2]: Martin Helmer, Anton Leykin, and Vidit Nanda. "Effective Whitney Stratification of Real Algebraic Varieties". Arxiv: 2307.05427.

i1 : S=QQ[x..z]

o1 = S

o1 : PolynomialRing

i2 : I=ideal(y^2*z-x^2)

            2     2
o2 = ideal(y z - x )

o2 : Ideal of S

i3 : polarVars I

                                                                      
o3 = {ideal (10163344z - 73462041, 5081672y - 72382095, 16200370336x -
     ------------------------------------------------------------------------
                                                                 2  
     620386936245), ideal (200154345987154y*z + 1747430111571076z  +
     ------------------------------------------------------------------------
                                                        2  
     77004596873425x - 258644393880625z, 22926102710641y  -
     ------------------------------------------------------------------------
                      2                                       
     1747430111571076z  - 154009193746850x + 258644393880625z,
     ------------------------------------------------------------------------
                                                           
     958363227262357797634x*y + 8366906813245811988196x*z +
     ------------------------------------------------------------------------
                             2                            
     28102913712083461939300z  + 1238420653872092055625x -
     ------------------------------------------------------------------------
                                      2     2
     4159629066255610515625z), ideal(y z - x )}

o3 : List

There are also several different options to preform the polar variety calculations. The default algorithm uses the M2 saturate command to compute the polar variteies, this option is Algorithm=>. The other options are: Algorithm=>"msolve" and Algorithm=>"M2F4". The Algorithm=>"msolve" only works in versions 1.25.06 and above of Macualay2. The Algorithm=>"M2F4" is mostly beneficial when working over a finite field. Note that over a finite field we can still sometimes obtain useful information, but the coefficients appearing in the resulting polynomials may (or likely will) be incorrect.

Ways to use polarVars:

polarVars(Ideal)

For the programmer

The object polarVars is a method function with options.

The source of this document is in WhitneyStratifications.m2:1799:0.