On differential polynomials
$begingroup$
Definition: A differential polynomial is a polynomial with indeterminates $y$, $y'$, $y''$, $ldots$ with coefficients in $K[x]$, algebra of polynomials with coefficients in a field $K$.
An example of a differential polynomial is
$$(1+x)y^2y'+(y'')^2(y''')^5-x^4 yy'y''y'''y^{(4)}-2x. $$
Such polynomials are studied in "Differential Algebraic Geometry" and solution of this polynomials are called "Differential Varieties".
My question: Is it possible to express a differential polynomial as an infinite series with terms of linear differential equations?
For example, is it possible to have an equality of the form
$$(y')^2=sum_{n=1}^{infty}a_n(x)y^{(n)}? $$
Of course, we need a norm to speak about convergence.
Note that the Taylor series of $(y')^2 $ does not satisfies the condition.
ordinary-differential-equations
asked Feb 24 '18 at 17:10
QurultayQurultay
601313
$endgroup$
add a comment |
$begingroup$
Definition: A differential polynomial is a polynomial with indeterminates $y$, $y'$, $y''$, $ldots$ with coefficients in $K[x]$, algebra of polynomials with coefficients in a field $K$.
An example of a differential polynomial is
$$(1+x)y^2y'+(y'')^2(y''')^5-x^4 yy'y''y'''y^{(4)}-2x. $$
Such polynomials are studied in "Differential Algebraic Geometry" and solution of this polynomials are called "Differential Varieties".
My question: Is it possible to express a differential polynomial as an infinite series with terms of linear differential equations?
For example, is it possible to have an equality of the form
$$(y')^2=sum_{n=1}^{infty}a_n(x)y^{(n)}? $$
Of course, we need a norm to speak about convergence.
Note that the Taylor series of $(y')^2 $ does not satisfies the condition.
ordinary-differential-equations
asked Feb 24 '18 at 17:10
QurultayQurultay
601313
$endgroup$
$begingroup$
In other words, you are asking whether there is an isomorphism of $K-$algebras $K[x,y,y',y'',dots] cong K[x][[y,y',y'',dots]] / (y,y',y''cdots)^2$ preserving the generators, right? I really do not think so, but I wouldn't be able to prove it on the spot. Why do you believe this is the case?
$endgroup$
– 57Jimmy
Feb 24 '18 at 17:25
$begingroup$
By the way, in your first example, I believe that the differential polynomial is either the RHS or the LHS, but not the equation itself
$endgroup$
– 57Jimmy
Feb 24 '18 at 17:28
$begingroup$
@57Jimmy Actually I don't believe that this is the case. But if it was true, then I would be unstuck in my research on differential varieties.
$endgroup$
– Qurultay
Feb 24 '18 at 17:29
$begingroup$
I see :) well, I am sorry that I cannot help you further, but the topic seems to be very interesting!
$endgroup$
– 57Jimmy
Feb 24 '18 at 17:30
$begingroup$
@57Jimmy Thanks, I edit it.
$endgroup$
– Qurultay
Feb 24 '18 at 17:30
add a comment |
$begingroup$
Definition: A differential polynomial is a polynomial with indeterminates $y$, $y'$, $y''$, $ldots$ with coefficients in $K[x]$, algebra of polynomials with coefficients in a field $K$.
An example of a differential polynomial is
$$(1+x)y^2y'+(y'')^2(y''')^5-x^4 yy'y''y'''y^{(4)}-2x. $$
Such polynomials are studied in "Differential Algebraic Geometry" and solution of this polynomials are called "Differential Varieties".
My question: Is it possible to express a differential polynomial as an infinite series with terms of linear differential equations?
For example, is it possible to have an equality of the form
$$(y')^2=sum_{n=1}^{infty}a_n(x)y^{(n)}? $$
Of course, we need a norm to speak about convergence.
Note that the Taylor series of $(y')^2 $ does not satisfies the condition.
ordinary-differential-equations
asked Feb 24 '18 at 17:10
QurultayQurultay
601313
$endgroup$
Definition: A differential polynomial is a polynomial with indeterminates $y$, $y'$, $y''$, $ldots$ with coefficients in $K[x]$, algebra of polynomials with coefficients in a field $K$.
An example of a differential polynomial is
$$(1+x)y^2y'+(y'')^2(y''')^5-x^4 yy'y''y'''y^{(4)}-2x. $$
Such polynomials are studied in "Differential Algebraic Geometry" and solution of this polynomials are called "Differential Varieties".
My question: Is it possible to express a differential polynomial as an infinite series with terms of linear differential equations?
For example, is it possible to have an equality of the form
$$(y')^2=sum_{n=1}^{infty}a_n(x)y^{(n)}? $$
Of course, we need a norm to speak about convergence.
Note that the Taylor series of $(y')^2 $ does not satisfies the condition.
ordinary-differential-equations
ordinary-differential-equations
asked Feb 24 '18 at 17:10
QurultayQurultay
601313
asked Feb 24 '18 at 17:10
QurultayQurultay
601313
edited Feb 24 '18 at 17:29
Qurultay
asked Feb 24 '18 at 17:10
QurultayQurultay
601313
asked Feb 24 '18 at 17:10
QurultayQurultay
601313
asked Feb 24 '18 at 17:10
QurultayQurultay
601313
601313
$begingroup$
In other words, you are asking whether there is an isomorphism of $K-$algebras $K[x,y,y',y'',dots] cong K[x][[y,y',y'',dots]] / (y,y',y''cdots)^2$ preserving the generators, right? I really do not think so, but I wouldn't be able to prove it on the spot. Why do you believe this is the case?
$endgroup$
– 57Jimmy
Feb 24 '18 at 17:25
$begingroup$
By the way, in your first example, I believe that the differential polynomial is either the RHS or the LHS, but not the equation itself
$endgroup$
– 57Jimmy
Feb 24 '18 at 17:28
$begingroup$
@57Jimmy Actually I don't believe that this is the case. But if it was true, then I would be unstuck in my research on differential varieties.
$endgroup$
– Qurultay
Feb 24 '18 at 17:29
$begingroup$
I see :) well, I am sorry that I cannot help you further, but the topic seems to be very interesting!
$endgroup$
– 57Jimmy
Feb 24 '18 at 17:30
$begingroup$
@57Jimmy Thanks, I edit it.
$endgroup$
– Qurultay
Feb 24 '18 at 17:30
add a comment |
$begingroup$
In other words, you are asking whether there is an isomorphism of $K-$algebras $K[x,y,y',y'',dots] cong K[x][[y,y',y'',dots]] / (y,y',y''cdots)^2$ preserving the generators, right? I really do not think so, but I wouldn't be able to prove it on the spot. Why do you believe this is the case?
$endgroup$
– 57Jimmy
Feb 24 '18 at 17:25
$begingroup$
By the way, in your first example, I believe that the differential polynomial is either the RHS or the LHS, but not the equation itself
$endgroup$
– 57Jimmy
Feb 24 '18 at 17:28
$begingroup$
@57Jimmy Actually I don't believe that this is the case. But if it was true, then I would be unstuck in my research on differential varieties.
$endgroup$
– Qurultay
Feb 24 '18 at 17:29
$begingroup$
I see :) well, I am sorry that I cannot help you further, but the topic seems to be very interesting!
$endgroup$
– 57Jimmy
Feb 24 '18 at 17:30
$begingroup$
@57Jimmy Thanks, I edit it.
$endgroup$
– Qurultay
Feb 24 '18 at 17:30
$begingroup$
In other words, you are asking whether there is an isomorphism of $K-$algebras $K[x,y,y',y'',dots] cong K[x][[y,y',y'',dots]] / (y,y',y''cdots)^2$ preserving the generators, right? I really do not think so, but I wouldn't be able to prove it on the spot. Why do you believe this is the case?
$endgroup$
– 57Jimmy
Feb 24 '18 at 17:25
$begingroup$
In other words, you are asking whether there is an isomorphism of $K-$algebras $K[x,y,y',y'',dots] cong K[x][[y,y',y'',dots]] / (y,y',y''cdots)^2$ preserving the generators, right? I really do not think so, but I wouldn't be able to prove it on the spot. Why do you believe this is the case?
$endgroup$
– 57Jimmy
Feb 24 '18 at 17:25
$begingroup$
By the way, in your first example, I believe that the differential polynomial is either the RHS or the LHS, but not the equation itself
$endgroup$
– 57Jimmy
Feb 24 '18 at 17:28
$begingroup$
By the way, in your first example, I believe that the differential polynomial is either the RHS or the LHS, but not the equation itself
$endgroup$
– 57Jimmy
Feb 24 '18 at 17:28
$begingroup$
@57Jimmy Actually I don't believe that this is the case. But if it was true, then I would be unstuck in my research on differential varieties.
$endgroup$
– Qurultay
Feb 24 '18 at 17:29
$begingroup$
@57Jimmy Actually I don't believe that this is the case. But if it was true, then I would be unstuck in my research on differential varieties.
$endgroup$
– Qurultay
Feb 24 '18 at 17:29
$begingroup$
I see :) well, I am sorry that I cannot help you further, but the topic seems to be very interesting!
$endgroup$
– 57Jimmy
Feb 24 '18 at 17:30
$begingroup$
I see :) well, I am sorry that I cannot help you further, but the topic seems to be very interesting!
$endgroup$
– 57Jimmy
Feb 24 '18 at 17:30
$begingroup$
@57Jimmy Thanks, I edit it.
$endgroup$
– Qurultay
Feb 24 '18 at 17:30
$begingroup$
@57Jimmy Thanks, I edit it.
$endgroup$
– Qurultay
Feb 24 '18 at 17:30
add a comment |
1 Answer
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I have an algebra $[E,F]=I, [J,E]=E, [J,F]=-F$, $I$ is central. I would like to have a differential polynomial realization of it, but such that it is not degenerate when $I$ acts as $0$. For example the dif. polynomial $E=isqrt{a} x$, $F=isqrt{a} partial_x$, $I=a$, $J=xpartial_x$ where $a$ is a number, is a realization, but $E=F=0$ when $a=0$. Is there another one such that neither $E$ nor $F$ is $0$ when $a=0$ ?
answered Jan 20 at 8:23
A.BabichenkoA.Babichenko
112
$endgroup$
$begingroup$
@A.Babichenko Actually, I do not understand what you mean. Please explain your question in details.
$endgroup$
– Qurultay
Jan 24 at 18:40
add a comment |
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1 Answer
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$begingroup$
I have an algebra $[E,F]=I, [J,E]=E, [J,F]=-F$, $I$ is central. I would like to have a differential polynomial realization of it, but such that it is not degenerate when $I$ acts as $0$. For example the dif. polynomial $E=isqrt{a} x$, $F=isqrt{a} partial_x$, $I=a$, $J=xpartial_x$ where $a$ is a number, is a realization, but $E=F=0$ when $a=0$. Is there another one such that neither $E$ nor $F$ is $0$ when $a=0$ ?
answered Jan 20 at 8:23
A.BabichenkoA.Babichenko
112
$endgroup$
$begingroup$
@A.Babichenko Actually, I do not understand what you mean. Please explain your question in details.
$endgroup$
– Qurultay
Jan 24 at 18:40
add a comment |
$begingroup$
I have an algebra $[E,F]=I, [J,E]=E, [J,F]=-F$, $I$ is central. I would like to have a differential polynomial realization of it, but such that it is not degenerate when $I$ acts as $0$. For example the dif. polynomial $E=isqrt{a} x$, $F=isqrt{a} partial_x$, $I=a$, $J=xpartial_x$ where $a$ is a number, is a realization, but $E=F=0$ when $a=0$. Is there another one such that neither $E$ nor $F$ is $0$ when $a=0$ ?
answered Jan 20 at 8:23
A.BabichenkoA.Babichenko
112
$endgroup$
$begingroup$
@A.Babichenko Actually, I do not understand what you mean. Please explain your question in details.
$endgroup$
– Qurultay
Jan 24 at 18:40
add a comment |
$begingroup$
I have an algebra $[E,F]=I, [J,E]=E, [J,F]=-F$, $I$ is central. I would like to have a differential polynomial realization of it, but such that it is not degenerate when $I$ acts as $0$. For example the dif. polynomial $E=isqrt{a} x$, $F=isqrt{a} partial_x$, $I=a$, $J=xpartial_x$ where $a$ is a number, is a realization, but $E=F=0$ when $a=0$. Is there another one such that neither $E$ nor $F$ is $0$ when $a=0$ ?
answered Jan 20 at 8:23
A.BabichenkoA.Babichenko
112
$endgroup$
I have an algebra $[E,F]=I, [J,E]=E, [J,F]=-F$, $I$ is central. I would like to have a differential polynomial realization of it, but such that it is not degenerate when $I$ acts as $0$. For example the dif. polynomial $E=isqrt{a} x$, $F=isqrt{a} partial_x$, $I=a$, $J=xpartial_x$ where $a$ is a number, is a realization, but $E=F=0$ when $a=0$. Is there another one such that neither $E$ nor $F$ is $0$ when $a=0$ ?
answered Jan 20 at 8:23
A.BabichenkoA.Babichenko
112
answered Jan 20 at 8:23
A.BabichenkoA.Babichenko
112
answered Jan 20 at 8:23
A.BabichenkoA.Babichenko
112
answered Jan 20 at 8:23
A.BabichenkoA.Babichenko
112
112
$begingroup$
@A.Babichenko Actually, I do not understand what you mean. Please explain your question in details.
$endgroup$
– Qurultay
Jan 24 at 18:40
add a comment |
$begingroup$
@A.Babichenko Actually, I do not understand what you mean. Please explain your question in details.
$endgroup$
– Qurultay
Jan 24 at 18:40
$begingroup$
@A.Babichenko Actually, I do not understand what you mean. Please explain your question in details.
$endgroup$
– Qurultay
Jan 24 at 18:40
$begingroup$
@A.Babichenko Actually, I do not understand what you mean. Please explain your question in details.
$endgroup$
– Qurultay
Jan 24 at 18:40
add a comment |
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$begingroup$
In other words, you are asking whether there is an isomorphism of $K-$algebras $K[x,y,y',y'',dots] cong K[x][[y,y',y'',dots]] / (y,y',y''cdots)^2$ preserving the generators, right? I really do not think so, but I wouldn't be able to prove it on the spot. Why do you believe this is the case?
$endgroup$
– 57Jimmy
Feb 24 '18 at 17:25
$begingroup$
By the way, in your first example, I believe that the differential polynomial is either the RHS or the LHS, but not the equation itself
$endgroup$
– 57Jimmy
Feb 24 '18 at 17:28
$begingroup$
@57Jimmy Actually I don't believe that this is the case. But if it was true, then I would be unstuck in my research on differential varieties.
$endgroup$
– Qurultay
Feb 24 '18 at 17:29
$begingroup$
I see :) well, I am sorry that I cannot help you further, but the topic seems to be very interesting!
$endgroup$
– 57Jimmy
Feb 24 '18 at 17:30
$begingroup$
@57Jimmy Thanks, I edit it.
$endgroup$
– Qurultay
Feb 24 '18 at 17:30