Linear polynomial whose multiplication with a given quadratic polynomial vanishes in the Jacobi ring
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 X
30911
add a comment |
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 X
30911
add a comment |
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 X
30911
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 X
30911
asked Dec 10 '18 at 21:03
User X
30911
asked Dec 10 '18 at 21:03
User X
30911
asked Dec 10 '18 at 21:03
User X
30911
asked Dec 10 '18 at 21:03
User X
30911
30911
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3034464%2flinear-polynomial-whose-multiplication-with-a-given-quadratic-polynomial-vanishe%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3034464%2flinear-polynomial-whose-multiplication-with-a-given-quadratic-polynomial-vanishe%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
