Finding the transformation that preserves the values of a polynomial.












2












$begingroup$


I have a polynomial in multiple variables, and would like to have a transformation act on them so that the value of the polynomial is preserved. For example, the polynomial $x^2+y^2$ is preserved in rotation about the origin. Is there a general way to obtain transformations that maintain the values of polynomials? Do these transformations always exist?



I am looking for specifically those transformations that form a nice, smooth, continuous group (just like rotation or the Lorentz transformation), and would like sufficiently many of them that any point may be transformed to any other point on its level set by their application.






polynomials transformation







share|cite|improve this question











$endgroup$












  • $begingroup$
    I guess you just need to study the level sets of whichever polynomial you're working with.
    $endgroup$
    – angryavian
    Dec 10 '18 at 20:57










  • $begingroup$
    The identity transformation always works. Do you have any conditions on the transformation? (Continuity, isometry, injectivity, etc.?) Otherwise you can construct weird transformations that keep points within their level sets...
    $endgroup$
    – angryavian
    Dec 10 '18 at 20:59










  • $begingroup$
    @angryavian What I am really looking for is a continuous group of transformations sufficiently large to transform any point on a level set to any other, thanks for reminding me that I needed to mention this...
    $endgroup$
    – Display Name
    Dec 10 '18 at 21:01










  • $begingroup$
    Do the transformations need to be linear on the vector $(x, y, ...)$?
    $endgroup$
    – Michael Seifert
    Dec 10 '18 at 21:05










  • $begingroup$
    @MichaelSeifert It would be nice if they were, but I think there might be some polynomials whose level sets linear transformations simply cannot preserve.
    $endgroup$
    – Display Name
    Dec 10 '18 at 21:06
















2












$begingroup$


I have a polynomial in multiple variables, and would like to have a transformation act on them so that the value of the polynomial is preserved. For example, the polynomial $x^2+y^2$ is preserved in rotation about the origin. Is there a general way to obtain transformations that maintain the values of polynomials? Do these transformations always exist?



I am looking for specifically those transformations that form a nice, smooth, continuous group (just like rotation or the Lorentz transformation), and would like sufficiently many of them that any point may be transformed to any other point on its level set by their application.






polynomials transformation







share|cite|improve this question











$endgroup$












  • $begingroup$
    I guess you just need to study the level sets of whichever polynomial you're working with.
    $endgroup$
    – angryavian
    Dec 10 '18 at 20:57










  • $begingroup$
    The identity transformation always works. Do you have any conditions on the transformation? (Continuity, isometry, injectivity, etc.?) Otherwise you can construct weird transformations that keep points within their level sets...
    $endgroup$
    – angryavian
    Dec 10 '18 at 20:59










  • $begingroup$
    @angryavian What I am really looking for is a continuous group of transformations sufficiently large to transform any point on a level set to any other, thanks for reminding me that I needed to mention this...
    $endgroup$
    – Display Name
    Dec 10 '18 at 21:01










  • $begingroup$
    Do the transformations need to be linear on the vector $(x, y, ...)$?
    $endgroup$
    – Michael Seifert
    Dec 10 '18 at 21:05










  • $begingroup$
    @MichaelSeifert It would be nice if they were, but I think there might be some polynomials whose level sets linear transformations simply cannot preserve.
    $endgroup$
    – Display Name
    Dec 10 '18 at 21:06














2












2








2


1



$begingroup$


I have a polynomial in multiple variables, and would like to have a transformation act on them so that the value of the polynomial is preserved. For example, the polynomial $x^2+y^2$ is preserved in rotation about the origin. Is there a general way to obtain transformations that maintain the values of polynomials? Do these transformations always exist?



I am looking for specifically those transformations that form a nice, smooth, continuous group (just like rotation or the Lorentz transformation), and would like sufficiently many of them that any point may be transformed to any other point on its level set by their application.






polynomials transformation







share|cite|improve this question











$endgroup$




I have a polynomial in multiple variables, and would like to have a transformation act on them so that the value of the polynomial is preserved. For example, the polynomial $x^2+y^2$ is preserved in rotation about the origin. Is there a general way to obtain transformations that maintain the values of polynomials? Do these transformations always exist?



I am looking for specifically those transformations that form a nice, smooth, continuous group (just like rotation or the Lorentz transformation), and would like sufficiently many of them that any point may be transformed to any other point on its level set by their application.






polynomials transformation




polynomials transformation






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 10 '18 at 21:02







Display Name

















asked Dec 10 '18 at 20:52









Display NameDisplay Name

2609




2609












  • $begingroup$
    I guess you just need to study the level sets of whichever polynomial you're working with.
    $endgroup$
    – angryavian
    Dec 10 '18 at 20:57










  • $begingroup$
    The identity transformation always works. Do you have any conditions on the transformation? (Continuity, isometry, injectivity, etc.?) Otherwise you can construct weird transformations that keep points within their level sets...
    $endgroup$
    – angryavian
    Dec 10 '18 at 20:59










  • $begingroup$
    @angryavian What I am really looking for is a continuous group of transformations sufficiently large to transform any point on a level set to any other, thanks for reminding me that I needed to mention this...
    $endgroup$
    – Display Name
    Dec 10 '18 at 21:01










  • $begingroup$
    Do the transformations need to be linear on the vector $(x, y, ...)$?
    $endgroup$
    – Michael Seifert
    Dec 10 '18 at 21:05










  • $begingroup$
    @MichaelSeifert It would be nice if they were, but I think there might be some polynomials whose level sets linear transformations simply cannot preserve.
    $endgroup$
    – Display Name
    Dec 10 '18 at 21:06


















  • $begingroup$
    I guess you just need to study the level sets of whichever polynomial you're working with.
    $endgroup$
    – angryavian
    Dec 10 '18 at 20:57










  • $begingroup$
    The identity transformation always works. Do you have any conditions on the transformation? (Continuity, isometry, injectivity, etc.?) Otherwise you can construct weird transformations that keep points within their level sets...
    $endgroup$
    – angryavian
    Dec 10 '18 at 20:59










  • $begingroup$
    @angryavian What I am really looking for is a continuous group of transformations sufficiently large to transform any point on a level set to any other, thanks for reminding me that I needed to mention this...
    $endgroup$
    – Display Name
    Dec 10 '18 at 21:01










  • $begingroup$
    Do the transformations need to be linear on the vector $(x, y, ...)$?
    $endgroup$
    – Michael Seifert
    Dec 10 '18 at 21:05










  • $begingroup$
    @MichaelSeifert It would be nice if they were, but I think there might be some polynomials whose level sets linear transformations simply cannot preserve.
    $endgroup$
    – Display Name
    Dec 10 '18 at 21:06
















$begingroup$
I guess you just need to study the level sets of whichever polynomial you're working with.
$endgroup$
– angryavian
Dec 10 '18 at 20:57




$begingroup$
I guess you just need to study the level sets of whichever polynomial you're working with.
$endgroup$
– angryavian
Dec 10 '18 at 20:57












$begingroup$
The identity transformation always works. Do you have any conditions on the transformation? (Continuity, isometry, injectivity, etc.?) Otherwise you can construct weird transformations that keep points within their level sets...
$endgroup$
– angryavian
Dec 10 '18 at 20:59




$begingroup$
The identity transformation always works. Do you have any conditions on the transformation? (Continuity, isometry, injectivity, etc.?) Otherwise you can construct weird transformations that keep points within their level sets...
$endgroup$
– angryavian
Dec 10 '18 at 20:59












$begingroup$
@angryavian What I am really looking for is a continuous group of transformations sufficiently large to transform any point on a level set to any other, thanks for reminding me that I needed to mention this...
$endgroup$
– Display Name
Dec 10 '18 at 21:01




$begingroup$
@angryavian What I am really looking for is a continuous group of transformations sufficiently large to transform any point on a level set to any other, thanks for reminding me that I needed to mention this...
$endgroup$
– Display Name
Dec 10 '18 at 21:01












$begingroup$
Do the transformations need to be linear on the vector $(x, y, ...)$?
$endgroup$
– Michael Seifert
Dec 10 '18 at 21:05




$begingroup$
Do the transformations need to be linear on the vector $(x, y, ...)$?
$endgroup$
– Michael Seifert
Dec 10 '18 at 21:05












$begingroup$
@MichaelSeifert It would be nice if they were, but I think there might be some polynomials whose level sets linear transformations simply cannot preserve.
$endgroup$
– Display Name
Dec 10 '18 at 21:06




$begingroup$
@MichaelSeifert It would be nice if they were, but I think there might be some polynomials whose level sets linear transformations simply cannot preserve.
$endgroup$
– Display Name
Dec 10 '18 at 21:06










1 Answer
1






active

oldest

votes


















1












$begingroup$

when the function is a quadratic form, meaning that, when we write the variables in a column vector $x,$ we have a symmetric matrix $H$ where the function is $x^T H x,$ the automorphism group is matrices $A$ such that
$$ A^T HA = H $$
It is typical to demand $det H neq 0,$ which gives $det A = pm 1.$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    The polynomials I would like to transform are sometimes higher than quadratic order, unfortunately. Although there might be some way to reduce them by adding more variables and more equations, like can be done with differential equations.
    $endgroup$
    – Display Name
    Dec 10 '18 at 21:12











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
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3034449%2ffinding-the-transformation-that-preserves-the-values-of-a-polynomial%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









1












$begingroup$

when the function is a quadratic form, meaning that, when we write the variables in a column vector $x,$ we have a symmetric matrix $H$ where the function is $x^T H x,$ the automorphism group is matrices $A$ such that
$$ A^T HA = H $$
It is typical to demand $det H neq 0,$ which gives $det A = pm 1.$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    The polynomials I would like to transform are sometimes higher than quadratic order, unfortunately. Although there might be some way to reduce them by adding more variables and more equations, like can be done with differential equations.
    $endgroup$
    – Display Name
    Dec 10 '18 at 21:12
















1












$begingroup$

when the function is a quadratic form, meaning that, when we write the variables in a column vector $x,$ we have a symmetric matrix $H$ where the function is $x^T H x,$ the automorphism group is matrices $A$ such that
$$ A^T HA = H $$
It is typical to demand $det H neq 0,$ which gives $det A = pm 1.$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    The polynomials I would like to transform are sometimes higher than quadratic order, unfortunately. Although there might be some way to reduce them by adding more variables and more equations, like can be done with differential equations.
    $endgroup$
    – Display Name
    Dec 10 '18 at 21:12














1












1








1





$begingroup$

when the function is a quadratic form, meaning that, when we write the variables in a column vector $x,$ we have a symmetric matrix $H$ where the function is $x^T H x,$ the automorphism group is matrices $A$ such that
$$ A^T HA = H $$
It is typical to demand $det H neq 0,$ which gives $det A = pm 1.$






share|cite|improve this answer









$endgroup$



when the function is a quadratic form, meaning that, when we write the variables in a column vector $x,$ we have a symmetric matrix $H$ where the function is $x^T H x,$ the automorphism group is matrices $A$ such that
$$ A^T HA = H $$
It is typical to demand $det H neq 0,$ which gives $det A = pm 1.$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 10 '18 at 21:06









Will JagyWill Jagy

103k5101200




103k5101200












  • $begingroup$
    The polynomials I would like to transform are sometimes higher than quadratic order, unfortunately. Although there might be some way to reduce them by adding more variables and more equations, like can be done with differential equations.
    $endgroup$
    – Display Name
    Dec 10 '18 at 21:12


















  • $begingroup$
    The polynomials I would like to transform are sometimes higher than quadratic order, unfortunately. Although there might be some way to reduce them by adding more variables and more equations, like can be done with differential equations.
    $endgroup$
    – Display Name
    Dec 10 '18 at 21:12
















$begingroup$
The polynomials I would like to transform are sometimes higher than quadratic order, unfortunately. Although there might be some way to reduce them by adding more variables and more equations, like can be done with differential equations.
$endgroup$
– Display Name
Dec 10 '18 at 21:12




$begingroup$
The polynomials I would like to transform are sometimes higher than quadratic order, unfortunately. Although there might be some way to reduce them by adding more variables and more equations, like can be done with differential equations.
$endgroup$
– Display Name
Dec 10 '18 at 21:12


















draft saved

draft discarded




















































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.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3034449%2ffinding-the-transformation-that-preserves-the-values-of-a-polynomial%23new-answer', 'question_page');
}
);

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







-

Popular posts from this blog

Probability when a professor distributes a quiz and homework assignment to a class of n students.

Image
0 0 $begingroup$ Need help with this problem. Suppose our lazy professor collects a quiz and a homework assignment from a class of n students one day, then distributes both the quizzes and the homework assignments back to the class in a random fashion for grading. Each student receives one quiz and one homework assignment to grade. (a) What is the probability that every student receives someone else's quiz to grade, and someone else's homework to grade? (b) What is the probability that no student receives both their own quiz and their own homework assignment to grade? In this case, some students may receive their own quiz, and others may receive their own homework assignment. (c) Compute the limiting probability as n approaches infinity in each case. ...
Read more

Aardman Animations

Image
Aardman Animations, Ltd. Aardman Animations Tipo privada Atividade Animação em stop-motion , Animação em CGI Fundação 1972 Fundador(es) Peter Lord David Sproxton Sede Bristol, Inglaterra  Reino Unido Proprietário(s) Dreamworks Pessoas-chave Peter Lord David Sproxton Nick Park Divisões Aardman Features Aardman Digital Aardman Commercials Aardman Broadcast Aardman International Aardman Rights Aardman Effects Aardman 3-D Systems Aardman Nathan Love AardBoiled Website oficial aardman.com ...
Read more

Are they similar matrix

Image
0 $begingroup$ Do $begin{bmatrix} 0&i&0\0&0&1\0&0&0 end{bmatrix} $ and $begin{bmatrix} 0&0&0\-i&0&0\0&1&0 end{bmatrix} $ are similar.Is this True/false Clearly both are nilpotent and one is conjucate transpose of other but how to know if they are similar.i'm stuck. Please help me linear-algebra vector-spaces eigenvalues-eigenvectors generalizedeigenvector share | cite | improve this question edited Dec 10 '18 at 6:38 Vasanth Kris asked Dec 10 '18 at 6:19 ...
Read more