Let $C_1$ and $C_2$ be the zeros of quadratic polynomials $f_1$ and $f_2$ respectively that don't have a...
$begingroup$
Artin Algebra Chapter 11
This has been answered here, but my questions are about the solution of Brian Bi:
(a) On proving g is the product of linear polynomials:
- It seems Brian Bi deduces the degree of $g$ in y is at most 2 because the degree of $g$ is at most 2. What's the exact relationship here? I think for polynomial $g(x,y)$ and $m ge 1$,
$$deg(g) ge max{deg(g) text{in y},deg(g) text{in x}}$$
Is that correct?
By the way, when we say a polynomial $p$ is any number of variables is linear (quadratic), does that imply $p$ is linear (quadratic) in each of the variables individually? Is the converse true (if it is not the case that the original statement is true by definition)?
It seems Brian Bi has cases for when the degree of $g$ in y is 1 or 2 but not 0. Can we have degree of $g$ in $y$ to be $0$?
If we can: If degree of $g$ in $y$ is $0$, so $g$ is a constant, then constant functions are linear so $g$ is a product of a linear polynomial namely itself?
If we cannot: why?
(b) On proving $|C_1 cap C_2| le 4$
Is $g$ a common factor of $f_1$ and $f_2$?
If so, why, and in the first place, why is $g$ a factor of either?
If not, what is the relevance of "don't have a common linear factor" ?
Is the $S_1$ and $S_2$ part implicitly using Exercise 11.9.4?
linear-algebra geometry algebraic-geometry polynomials ring-theory
$endgroup$
add a comment |
$begingroup$
Artin Algebra Chapter 11
This has been answered here, but my questions are about the solution of Brian Bi:
(a) On proving g is the product of linear polynomials:
- It seems Brian Bi deduces the degree of $g$ in y is at most 2 because the degree of $g$ is at most 2. What's the exact relationship here? I think for polynomial $g(x,y)$ and $m ge 1$,
$$deg(g) ge max{deg(g) text{in y},deg(g) text{in x}}$$
Is that correct?
By the way, when we say a polynomial $p$ is any number of variables is linear (quadratic), does that imply $p$ is linear (quadratic) in each of the variables individually? Is the converse true (if it is not the case that the original statement is true by definition)?
It seems Brian Bi has cases for when the degree of $g$ in y is 1 or 2 but not 0. Can we have degree of $g$ in $y$ to be $0$?
If we can: If degree of $g$ in $y$ is $0$, so $g$ is a constant, then constant functions are linear so $g$ is a product of a linear polynomial namely itself?
If we cannot: why?
(b) On proving $|C_1 cap C_2| le 4$
Is $g$ a common factor of $f_1$ and $f_2$?
If so, why, and in the first place, why is $g$ a factor of either?
If not, what is the relevance of "don't have a common linear factor" ?
Is the $S_1$ and $S_2$ part implicitly using Exercise 11.9.4?
linear-algebra geometry algebraic-geometry polynomials ring-theory
$endgroup$
add a comment |
$begingroup$
Artin Algebra Chapter 11
This has been answered here, but my questions are about the solution of Brian Bi:
(a) On proving g is the product of linear polynomials:
- It seems Brian Bi deduces the degree of $g$ in y is at most 2 because the degree of $g$ is at most 2. What's the exact relationship here? I think for polynomial $g(x,y)$ and $m ge 1$,
$$deg(g) ge max{deg(g) text{in y},deg(g) text{in x}}$$
Is that correct?
By the way, when we say a polynomial $p$ is any number of variables is linear (quadratic), does that imply $p$ is linear (quadratic) in each of the variables individually? Is the converse true (if it is not the case that the original statement is true by definition)?
It seems Brian Bi has cases for when the degree of $g$ in y is 1 or 2 but not 0. Can we have degree of $g$ in $y$ to be $0$?
If we can: If degree of $g$ in $y$ is $0$, so $g$ is a constant, then constant functions are linear so $g$ is a product of a linear polynomial namely itself?
If we cannot: why?
(b) On proving $|C_1 cap C_2| le 4$
Is $g$ a common factor of $f_1$ and $f_2$?
If so, why, and in the first place, why is $g$ a factor of either?
If not, what is the relevance of "don't have a common linear factor" ?
Is the $S_1$ and $S_2$ part implicitly using Exercise 11.9.4?
linear-algebra geometry algebraic-geometry polynomials ring-theory
$endgroup$
Artin Algebra Chapter 11
This has been answered here, but my questions are about the solution of Brian Bi:
(a) On proving g is the product of linear polynomials:
- It seems Brian Bi deduces the degree of $g$ in y is at most 2 because the degree of $g$ is at most 2. What's the exact relationship here? I think for polynomial $g(x,y)$ and $m ge 1$,
$$deg(g) ge max{deg(g) text{in y},deg(g) text{in x}}$$
Is that correct?
By the way, when we say a polynomial $p$ is any number of variables is linear (quadratic), does that imply $p$ is linear (quadratic) in each of the variables individually? Is the converse true (if it is not the case that the original statement is true by definition)?
It seems Brian Bi has cases for when the degree of $g$ in y is 1 or 2 but not 0. Can we have degree of $g$ in $y$ to be $0$?
If we can: If degree of $g$ in $y$ is $0$, so $g$ is a constant, then constant functions are linear so $g$ is a product of a linear polynomial namely itself?
If we cannot: why?
(b) On proving $|C_1 cap C_2| le 4$
Is $g$ a common factor of $f_1$ and $f_2$?
If so, why, and in the first place, why is $g$ a factor of either?
If not, what is the relevance of "don't have a common linear factor" ?
Is the $S_1$ and $S_2$ part implicitly using Exercise 11.9.4?
linear-algebra geometry algebraic-geometry polynomials ring-theory
linear-algebra geometry algebraic-geometry polynomials ring-theory
edited Dec 30 '18 at 11:14
asked Dec 30 '18 at 11:08
user198044
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%2f3056722%2flet-c-1-and-c-2-be-the-zeros-of-quadratic-polynomials-f-1-and-f-2-respec%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.
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%2f3056722%2flet-c-1-and-c-2-be-the-zeros-of-quadratic-polynomials-f-1-and-f-2-respec%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
