Linear polynomial whose multiplication with a given quadratic polynomial vanishes in the Jacobi ring
$begingroup$
Let $mathbb C[x_1,ldots,x_n]$ be the polynomial ring of $n$ varibles, and $mathbb C[x_1,ldots,x_n]_d$ be its homogeneous degree $d$ piece. Let $Fin mathbb C[x_1,ldots,x_n]_3$ be a homogeneous cubic polynomial which has smooth zero locus. Then
$$P=mathbb C{x_i partial_j F}$$
is a linear space of dimension $n^2$ (by smoothness). For any homogeneous quadratic polynomial $q in mathbb C[x_1,ldots,x_n]_2$, we define
$$n(q)=dim{fin mathbb C[x_1,ldots,x_n]_1: fcdot q in P }$$
which can be also formulated as the dimension of the set of linear polynomial $f$ whose multiplication with $q$ vanishes in the Jacobi ring.
For example, easy to see $n(q)=n$ if and only if $q=sum c_i partial_i F$.
I computed several examples, and it seems the following is always hold:
If $qneq sum c_i partial_i F$, then $n(q)leq2$.
But I don't know how to prove it. I can show it in some special case, but in general ${partial_iF}$ is quite mysterious to me. Also it seems the function $n(q)$ is hard to control.
Could someone help me? Thanks in advance!
abstract-algebra algebraic-geometry polynomials homogeneous-equation
asked Dec 10 '18 at 21:03
User XUser X
32911
$endgroup$
add a comment |
$begingroup$
Let $mathbb C[x_1,ldots,x_n]$ be the polynomial ring of $n$ varibles, and $mathbb C[x_1,ldots,x_n]_d$ be its homogeneous degree $d$ piece. Let $Fin mathbb C[x_1,ldots,x_n]_3$ be a homogeneous cubic polynomial which has smooth zero locus. Then
$$P=mathbb C{x_i partial_j F}$$
is a linear space of dimension $n^2$ (by smoothness). For any homogeneous quadratic polynomial $q in mathbb C[x_1,ldots,x_n]_2$, we define
$$n(q)=dim{fin mathbb C[x_1,ldots,x_n]_1: fcdot q in P }$$
which can be also formulated as the dimension of the set of linear polynomial $f$ whose multiplication with $q$ vanishes in the Jacobi ring.
For example, easy to see $n(q)=n$ if and only if $q=sum c_i partial_i F$.
I computed several examples, and it seems the following is always hold:
If $qneq sum c_i partial_i F$, then $n(q)leq2$.
But I don't know how to prove it. I can show it in some special case, but in general ${partial_iF}$ is quite mysterious to me. Also it seems the function $n(q)$ is hard to control.
Could someone help me? Thanks in advance!
abstract-algebra algebraic-geometry polynomials homogeneous-equation
asked Dec 10 '18 at 21:03
User XUser X
32911
$endgroup$
add a comment |
$begingroup$
Let $mathbb C[x_1,ldots,x_n]$ be the polynomial ring of $n$ varibles, and $mathbb C[x_1,ldots,x_n]_d$ be its homogeneous degree $d$ piece. Let $Fin mathbb C[x_1,ldots,x_n]_3$ be a homogeneous cubic polynomial which has smooth zero locus. Then
$$P=mathbb C{x_i partial_j F}$$
is a linear space of dimension $n^2$ (by smoothness). For any homogeneous quadratic polynomial $q in mathbb C[x_1,ldots,x_n]_2$, we define
$$n(q)=dim{fin mathbb C[x_1,ldots,x_n]_1: fcdot q in P }$$
which can be also formulated as the dimension of the set of linear polynomial $f$ whose multiplication with $q$ vanishes in the Jacobi ring.
For example, easy to see $n(q)=n$ if and only if $q=sum c_i partial_i F$.
I computed several examples, and it seems the following is always hold:
If $qneq sum c_i partial_i F$, then $n(q)leq2$.
But I don't know how to prove it. I can show it in some special case, but in general ${partial_iF}$ is quite mysterious to me. Also it seems the function $n(q)$ is hard to control.
Could someone help me? Thanks in advance!
abstract-algebra algebraic-geometry polynomials homogeneous-equation
asked Dec 10 '18 at 21:03
User XUser X
32911
$endgroup$
Let $mathbb C[x_1,ldots,x_n]$ be the polynomial ring of $n$ varibles, and $mathbb C[x_1,ldots,x_n]_d$ be its homogeneous degree $d$ piece. Let $Fin mathbb C[x_1,ldots,x_n]_3$ be a homogeneous cubic polynomial which has smooth zero locus. Then
$$P=mathbb C{x_i partial_j F}$$
is a linear space of dimension $n^2$ (by smoothness). For any homogeneous quadratic polynomial $q in mathbb C[x_1,ldots,x_n]_2$, we define
$$n(q)=dim{fin mathbb C[x_1,ldots,x_n]_1: fcdot q in P }$$
which can be also formulated as the dimension of the set of linear polynomial $f$ whose multiplication with $q$ vanishes in the Jacobi ring.
For example, easy to see $n(q)=n$ if and only if $q=sum c_i partial_i F$.
I computed several examples, and it seems the following is always hold:
If $qneq sum c_i partial_i F$, then $n(q)leq2$.
But I don't know how to prove it. I can show it in some special case, but in general ${partial_iF}$ is quite mysterious to me. Also it seems the function $n(q)$ is hard to control.
Could someone help me? Thanks in advance!
abstract-algebra algebraic-geometry polynomials homogeneous-equation
abstract-algebra algebraic-geometry polynomials homogeneous-equation
asked Dec 10 '18 at 21:03
User XUser X
32911
asked Dec 10 '18 at 21:03
User XUser X
32911
asked Dec 10 '18 at 21:03
User XUser X
32911
asked Dec 10 '18 at 21:03
User XUser X
32911
asked Dec 10 '18 at 21:03
User XUser X
32911
32911
add a comment |
add a comment |
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