Computing singular locus

asked 2023-02-24 14:43:02 +0100

Jesus Martinez Garcia gravatar image

updated 2023-02-24 18:07:16 +0100

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:

$$F=a_0x_0f_3(x_1, x_2, x_3)+x_1^2f_2(x_0, \ldots, x_4)+x_0x_1g_3(x_0, x_1,x_2,x_3,x_4)$$

where $f_i, g_j$ 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 $x_1, x_2, x_3$ with arbitrary coefficients $a_1, a_2, \ldots$

I don't care much which field the coefficients belong to (but if you must know, let it be $\mathbb C$). I want to find the singular locus (i.e. the points $p\in \mathbb P^4$ where all partial derivatives $\frac{\partial F}{\partial x_i}(p)$ of $F$ simultaneously vanish) of one such $F$ in terms of symbolic coefficients and variables $x_i$. 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. $F=a_0x_0x_1^3+a_1x_0x_1x_2^2+\cdots$).

Thank you.

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Comments

1

By "singular", do you mean

  • points where all derivatves of $F$ wrt to $(x_0,x_1\dots,x_4)$ 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 ( 2023-02-24 15:05:36 +0100 )edit
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 ( 2023-02-24 16:31:33 +0100 )edit

@Emmanuel Charpentier Thanks for your question. I mean the first option. I have edited accordingly. My apologies..

Jesus Martinez Garcia gravatar imageJesus Martinez Garcia ( 2023-02-24 18:08:13 +0100 )edit

@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 ( 2023-02-24 18:12:27 +0100 )edit
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 ( 2023-02-24 19:22:22 +0100 )edit
see more comments