Computing singular locus
 
  
 
  
 
 
 
 
 
 
    
        
        asked 2 years ago  
 
 Jesus Martinez Garcia gravatar image  
 
 
 
 
    
        
        updated 2 years ago  
 
 
 
 
 
I have a bunch of homogeneous polynomials in 5 variables with specific arbitrary (symbolic) non-zero coefficients, i.e. some of them are zero and some of them are non-zero but I don't know the value. For instance:
 
F=a0x0f3(x1,x2,x3)+x21f2(x0,…,x4)+x0x1g3(x0,x1,x2,x3,x4)
 
where fi,gj are the polynomials with arbitrary non-zero coefficients of degree i, j, respectively, e.g. say that f_3 has all possible monomials of degree 3 in variables x1,x2,x3 with arbitrary coefficients a1,a2,…
 
I don't care much which field the coefficients belong to (but if you must know, let it be C). I want to find the singular locus (i.e. the points p∈P4 where all partial derivatives ∂F∂xi(p) of F simultaneously vanish) of one such F in terms of symbolic coefficients and variables xi. Is this something Sagemath can do? If so, can you give me a MWE? For the avoidance of doubt, I am happy to rewrite F above to make the coefficients explicit (e.g. F=a0x0x31+a1x0x1x22+⋯).
 
Thank you.
 
Preview: (hide)
 
 
 
 
          
 
Comments
1
By "singular", do you mean
  • points where all derivatves of F wrt to (x0,x1…,x4) simultaneously vanish to 0 ?
  • or possibly points where two or more, but not necessarily all five, of those derivatives vanish ?
  • or something else ?
Emmanuel Charpentier gravatar imageEmmanuel Charpentier (
    
        2 years ago
    )
1
In https://www.singular.uni-kl.de/ftp/pu...  there are some examples of investigating the singular locus. Singular is a part of Sage.
In   https://faculty.math.illinois.edu/Mac...   one can find some examples of finding  the singular locus with  Macaulay 2  which can be used in Cocalc  and Sage CellServer.
Singular locus is also mentioned in 
https://doc.sagemath.org/html/en/refe...   and  https://trac.sagemath.org/ticket/3253
achrzesz gravatar imageachrzesz (
    
        2 years ago
    )
@Emmanuel Charpentier Thanks for your question. I mean the first option. I have edited accordingly. My apologies..
Jesus Martinez Garcia gravatar imageJesus Martinez Garcia (
    
        2 years ago
    )
@achrzesz Thanks for your message. Is there a way to define in 
https://doc.sagemath.org/html/en/refe...z%5E2%20%2D%204x*z%5E3%20%2D
the ideal with arbitrary coefficients instead of specific ones? The problem I find is that I don't want the coefficients to be treated as variables but as 'arbitrary non-zero constants'.
Jesus Martinez Garcia gravatar imageJesus Martinez Garcia (
    
        2 years ago
    )
1
Do you mean
R=PolynomialRing(SR,'x',5)
x=R.gens()
a=var('a',n=5)
p=a0*x[0]^2+a2*x[2]^3
J=p.jacobian_ideal();J
Note however:
J.dimension()
verbose 0 (4043: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation.
verbose 0 (1131: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.

3
(For a better solution see more comments)
achrzesz gravatar imageachrzesz (
    
        2 years ago
    )

            
                see more comments