Finding the transformation that preserves the values of a polynomial.
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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
asked Dec 10 '18 at 20:52
Display NameDisplay Name
2609
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add a comment |
$begingroup$
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
asked Dec 10 '18 at 20:52
Display NameDisplay Name
2609
$endgroup$
$begingroup$
I guess you just need to study the level sets of whichever polynomial you're working with.
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– angryavian
Dec 10 '18 at 20:57
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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...
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– angryavian
Dec 10 '18 at 20:59
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@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...
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– Display Name
Dec 10 '18 at 21:01
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Do the transformations need to be linear on the vector $(x, y, ...)$?
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– Michael Seifert
Dec 10 '18 at 21:05
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@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
add a comment |
$begingroup$
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
asked Dec 10 '18 at 20:52
Display NameDisplay Name
2609
$endgroup$
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
asked Dec 10 '18 at 20:52
Display NameDisplay Name
2609
asked Dec 10 '18 at 20:52
Display NameDisplay Name
2609
edited Dec 10 '18 at 21:02
Display Name
asked Dec 10 '18 at 20:52
Display NameDisplay Name
2609
asked Dec 10 '18 at 20:52
Display NameDisplay Name
2609
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
add a comment |
$begingroup$
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
add a comment |
1 Answer
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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 $det A = pm 1.$
answered Dec 10 '18 at 21:06
Will JagyWill Jagy
103k5101200
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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.
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– Display Name
Dec 10 '18 at 21:12
add a comment |
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1 Answer
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1 Answer
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$begingroup$
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 $det A = pm 1.$
answered Dec 10 '18 at 21:06
Will JagyWill Jagy
103k5101200
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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 $det A = pm 1.$
answered Dec 10 '18 at 21:06
Will JagyWill Jagy
103k5101200
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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 $det A = pm 1.$
answered Dec 10 '18 at 21:06
Will JagyWill Jagy
103k5101200
$endgroup$
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 $det A = pm 1.$
answered Dec 10 '18 at 21:06
Will JagyWill Jagy
103k5101200
answered Dec 10 '18 at 21:06
Will JagyWill Jagy
103k5101200
answered Dec 10 '18 at 21:06
Will JagyWill Jagy
103k5101200
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
add a comment |
$begingroup$
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
add a comment |
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$begingroup$
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