The first principal subresultant coefficient of two polynomials
$begingroup$
Let $f=f(x), g=g(x) in mathbb{C}[x]$, with $deg(f)=deg(g)=n geq 2$.
Write $f=(x-a_1)cdots(x-a_n)$ and $g=(x-b_1)cdots(x-b_n)$,
where $a_i,b_i in mathbb{C}$, $1 leq i leq n$.
Let $lambda,mu in mathbb{C}$.
Question 1: Is it true that the first principal subresultant coefficient, $s_1$ in wikipedia notations, of $f-lambda$ and $g-mu$ equals $(a_1+cdots+a_n)-(b_1+cdots+b_n)$?
Notice that this expression is independent of $lambda$ and $mu$.
I have played with WolframAlpha and for $n in {2,3}$ the answer is positive; however, I am afraid that I had an error, and the $n=3$ case does not have a positive answer? (The $n=2$ case seems to have a positive answer).
Question 2: Could one please attach the exact form of $s_1$ when $n=3$?
More generally, if $deg(f)=n$ and $deg(g)=m$ with $n,m geq 2$ not necessarily equal:
Question 3: What is the exact definition of $s_1$? Can it be easily computed for $prod_{i=1}^{n} (x-a_i) - lambda$ and $prod_{j=1}^{m} (x-b_j) - mu$?
Notice that it may not be true that the first principal subresultant coefficient $s_1$ is as above (sum of the $a_i$'a minus sum of the $b_i$'s), see the answer to this question, which (among other things) says the following:
Avoiding tedious hand calculations, the subresultants of $x^3-4x-lambda$ and $x^2+1-mu$ courtesy WA subresultants[ x^3 - 4x - lambda , x^2 + 1 - mu , x ] are:
$$
begin{align}
s_0 &= λ^2 - μ^3 + 11 μ^2 - 35 μ + 25 \
s_1 &= μ - 5 \
s_2 &= 1
end{align}
$$
Thank you very much!
polynomials commutative-algebra wolfram-alpha resultant
asked Dec 29 '18 at 22:22
user237522user237522
2,1651617
$endgroup$
add a comment |
$begingroup$
Let $f=f(x), g=g(x) in mathbb{C}[x]$, with $deg(f)=deg(g)=n geq 2$.
Write $f=(x-a_1)cdots(x-a_n)$ and $g=(x-b_1)cdots(x-b_n)$,
where $a_i,b_i in mathbb{C}$, $1 leq i leq n$.
Let $lambda,mu in mathbb{C}$.
Question 1: Is it true that the first principal subresultant coefficient, $s_1$ in wikipedia notations, of $f-lambda$ and $g-mu$ equals $(a_1+cdots+a_n)-(b_1+cdots+b_n)$?
Notice that this expression is independent of $lambda$ and $mu$.
I have played with WolframAlpha and for $n in {2,3}$ the answer is positive; however, I am afraid that I had an error, and the $n=3$ case does not have a positive answer? (The $n=2$ case seems to have a positive answer).
Question 2: Could one please attach the exact form of $s_1$ when $n=3$?
More generally, if $deg(f)=n$ and $deg(g)=m$ with $n,m geq 2$ not necessarily equal:
Question 3: What is the exact definition of $s_1$? Can it be easily computed for $prod_{i=1}^{n} (x-a_i) - lambda$ and $prod_{j=1}^{m} (x-b_j) - mu$?
Notice that it may not be true that the first principal subresultant coefficient $s_1$ is as above (sum of the $a_i$'a minus sum of the $b_i$'s), see the answer to this question, which (among other things) says the following:
Avoiding tedious hand calculations, the subresultants of $x^3-4x-lambda$ and $x^2+1-mu$ courtesy WA subresultants[ x^3 - 4x - lambda , x^2 + 1 - mu , x ] are:
$$
begin{align}
s_0 &= λ^2 - μ^3 + 11 μ^2 - 35 μ + 25 \
s_1 &= μ - 5 \
s_2 &= 1
end{align}
$$
Thank you very much!
polynomials commutative-algebra wolfram-alpha resultant
asked Dec 29 '18 at 22:22
user237522user237522
2,1651617
$endgroup$
$begingroup$
An answer to Question 2 can be found in the following link: mathematica.stackexchange.com/questions/188570/… Unfortunately, it shows that for degree $n=3$, we have $s_2=a_1+a_2+a_3-b_1-b_2-b_3$, not $s_1=a_1+a_2+a_3-b_1-b_2-b_3$...
$endgroup$
– user237522
Dec 30 '18 at 1:40
add a comment |
$begingroup$
Let $f=f(x), g=g(x) in mathbb{C}[x]$, with $deg(f)=deg(g)=n geq 2$.
Write $f=(x-a_1)cdots(x-a_n)$ and $g=(x-b_1)cdots(x-b_n)$,
where $a_i,b_i in mathbb{C}$, $1 leq i leq n$.
Let $lambda,mu in mathbb{C}$.
Question 1: Is it true that the first principal subresultant coefficient, $s_1$ in wikipedia notations, of $f-lambda$ and $g-mu$ equals $(a_1+cdots+a_n)-(b_1+cdots+b_n)$?
Notice that this expression is independent of $lambda$ and $mu$.
I have played with WolframAlpha and for $n in {2,3}$ the answer is positive; however, I am afraid that I had an error, and the $n=3$ case does not have a positive answer? (The $n=2$ case seems to have a positive answer).
Question 2: Could one please attach the exact form of $s_1$ when $n=3$?
More generally, if $deg(f)=n$ and $deg(g)=m$ with $n,m geq 2$ not necessarily equal:
Question 3: What is the exact definition of $s_1$? Can it be easily computed for $prod_{i=1}^{n} (x-a_i) - lambda$ and $prod_{j=1}^{m} (x-b_j) - mu$?
Notice that it may not be true that the first principal subresultant coefficient $s_1$ is as above (sum of the $a_i$'a minus sum of the $b_i$'s), see the answer to this question, which (among other things) says the following:
Avoiding tedious hand calculations, the subresultants of $x^3-4x-lambda$ and $x^2+1-mu$ courtesy WA subresultants[ x^3 - 4x - lambda , x^2 + 1 - mu , x ] are:
$$
begin{align}
s_0 &= λ^2 - μ^3 + 11 μ^2 - 35 μ + 25 \
s_1 &= μ - 5 \
s_2 &= 1
end{align}
$$
Thank you very much!
polynomials commutative-algebra wolfram-alpha resultant
asked Dec 29 '18 at 22:22
user237522user237522
2,1651617
$endgroup$
Let $f=f(x), g=g(x) in mathbb{C}[x]$, with $deg(f)=deg(g)=n geq 2$.
Write $f=(x-a_1)cdots(x-a_n)$ and $g=(x-b_1)cdots(x-b_n)$,
where $a_i,b_i in mathbb{C}$, $1 leq i leq n$.
Let $lambda,mu in mathbb{C}$.
Question 1: Is it true that the first principal subresultant coefficient, $s_1$ in wikipedia notations, of $f-lambda$ and $g-mu$ equals $(a_1+cdots+a_n)-(b_1+cdots+b_n)$?
Notice that this expression is independent of $lambda$ and $mu$.
I have played with WolframAlpha and for $n in {2,3}$ the answer is positive; however, I am afraid that I had an error, and the $n=3$ case does not have a positive answer? (The $n=2$ case seems to have a positive answer).
Question 2: Could one please attach the exact form of $s_1$ when $n=3$?
More generally, if $deg(f)=n$ and $deg(g)=m$ with $n,m geq 2$ not necessarily equal:
Question 3: What is the exact definition of $s_1$? Can it be easily computed for $prod_{i=1}^{n} (x-a_i) - lambda$ and $prod_{j=1}^{m} (x-b_j) - mu$?
Notice that it may not be true that the first principal subresultant coefficient $s_1$ is as above (sum of the $a_i$'a minus sum of the $b_i$'s), see the answer to this question, which (among other things) says the following:
Avoiding tedious hand calculations, the subresultants of $x^3-4x-lambda$ and $x^2+1-mu$ courtesy WA subresultants[ x^3 - 4x - lambda , x^2 + 1 - mu , x ] are:
$$
begin{align}
s_0 &= λ^2 - μ^3 + 11 μ^2 - 35 μ + 25 \
s_1 &= μ - 5 \
s_2 &= 1
end{align}
$$
Thank you very much!
polynomials commutative-algebra wolfram-alpha resultant
polynomials commutative-algebra wolfram-alpha resultant
asked Dec 29 '18 at 22:22
user237522user237522
2,1651617
asked Dec 29 '18 at 22:22
user237522user237522
2,1651617
edited Dec 30 '18 at 0:43
user237522
asked Dec 29 '18 at 22:22
user237522user237522
2,1651617
asked Dec 29 '18 at 22:22
user237522user237522
2,1651617
asked Dec 29 '18 at 22:22
user237522user237522
2,1651617
2,1651617
$begingroup$
An answer to Question 2 can be found in the following link: mathematica.stackexchange.com/questions/188570/… Unfortunately, it shows that for degree $n=3$, we have $s_2=a_1+a_2+a_3-b_1-b_2-b_3$, not $s_1=a_1+a_2+a_3-b_1-b_2-b_3$...
$endgroup$
– user237522
Dec 30 '18 at 1:40
add a comment |
$begingroup$
An answer to Question 2 can be found in the following link: mathematica.stackexchange.com/questions/188570/… Unfortunately, it shows that for degree $n=3$, we have $s_2=a_1+a_2+a_3-b_1-b_2-b_3$, not $s_1=a_1+a_2+a_3-b_1-b_2-b_3$...
$endgroup$
– user237522
Dec 30 '18 at 1:40
$begingroup$
An answer to Question 2 can be found in the following link: mathematica.stackexchange.com/questions/188570/… Unfortunately, it shows that for degree $n=3$, we have $s_2=a_1+a_2+a_3-b_1-b_2-b_3$, not $s_1=a_1+a_2+a_3-b_1-b_2-b_3$...
$endgroup$
– user237522
Dec 30 '18 at 1:40
$begingroup$
An answer to Question 2 can be found in the following link: mathematica.stackexchange.com/questions/188570/… Unfortunately, it shows that for degree $n=3$, we have $s_2=a_1+a_2+a_3-b_1-b_2-b_3$, not $s_1=a_1+a_2+a_3-b_1-b_2-b_3$...
$endgroup$
– user237522
Dec 30 '18 at 1:40
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%2f3056313%2fthe-first-principal-subresultant-coefficient-of-two-polynomials%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%2f3056313%2fthe-first-principal-subresultant-coefficient-of-two-polynomials%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
$begingroup$
An answer to Question 2 can be found in the following link: mathematica.stackexchange.com/questions/188570/… Unfortunately, it shows that for degree $n=3$, we have $s_2=a_1+a_2+a_3-b_1-b_2-b_3$, not $s_1=a_1+a_2+a_3-b_1-b_2-b_3$...
$endgroup$
– user237522
Dec 30 '18 at 1:40