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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 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.

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 $\frac{\partial F}{\partial x_i} of$F$ 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.

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 $\frac{\partial F}{\partial x_i} x_i}$ of $F$ 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.

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 $\frac{\partial F}{\partial x_i}$ 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.