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Explain Application of Risch's Structure Theorem for Elementary Functions
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Could you please explain the application of Risch's structure theorem for elementary functions and give some detailed examples?
The Structure Theorem for Elementary Functions:
"Let $(mathfrak{E},Y)$ be elementary over $(mathfrak{D},X)$. kernel $Y= K$. $mathfrak{D}_1=mathfrak{D}(theta_1,...,theta_m)$, $mathfrak{E}=mathfrak{D}_1(theta)$. Let $z_1=e^{y_1},...,z_q=e^{y_q},log z_{q+1}=y_{q+1},...,log z_r=y_r$ be the exponentials and logarithms occurring among $theta_1,...,theta_m$. (See remark following definition at beginning of this part.) Suppose either $theta=z=e^y$ is algebraic over $mathfrak{D}_1$, or $theta=y=log z$ is algebraic over $mathfrak{D}_1$ (where $y$ or respectively $z$ is in $mathfrak{D}_1$). Then
(1) there are $c_iin K$, $fin tilde{mathfrak{D}}$ (= alg. closure of $mathfrak{D}$ in $mathfrak{E}$) such that $y+sum_1^rc_iy_i=f$, and (after a permutation of indices) $1,c_{s+1},...,c_r,0le sle r$ form a
maximal $mathbb{Q}$ linearly independent set of the coefficients $1,c_1,...,c_r$;
(2) there are $nneq 0$, $n_iinmathbb{Z}$, $ginmathfrak{D}$ such that $z^nPi_{i=1}^s z_i^{n_i}=g$.
Furthermore if $(mathfrak{D},X)=(K(z),d/dz)$, then $f$ and $g$ can be chosen to be in $K$, $r=s$, and $c_i=n_i/n$, $i=1,...,r$."
[Risch 1979] Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math 101 (1979) (4) 743-759
I know the mathematical terms field, differential field, field extension, elementary extension, elementary function, and all the mathematical terms in the theorem.
But how can the theorem help in detail to decide if a given $theta$ is algebraic over $mathfrak{D}_1$?
abstract-algebra differential-algebra
asked Jan 27 at 15:51
IV_IV_
1,556525
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add a comment |
$begingroup$
Could you please explain the application of Risch's structure theorem for elementary functions and give some detailed examples?
The Structure Theorem for Elementary Functions:
"Let $(mathfrak{E},Y)$ be elementary over $(mathfrak{D},X)$. kernel $Y= K$. $mathfrak{D}_1=mathfrak{D}(theta_1,...,theta_m)$, $mathfrak{E}=mathfrak{D}_1(theta)$. Let $z_1=e^{y_1},...,z_q=e^{y_q},log z_{q+1}=y_{q+1},...,log z_r=y_r$ be the exponentials and logarithms occurring among $theta_1,...,theta_m$. (See remark following definition at beginning of this part.) Suppose either $theta=z=e^y$ is algebraic over $mathfrak{D}_1$, or $theta=y=log z$ is algebraic over $mathfrak{D}_1$ (where $y$ or respectively $z$ is in $mathfrak{D}_1$). Then
(1) there are $c_iin K$, $fin tilde{mathfrak{D}}$ (= alg. closure of $mathfrak{D}$ in $mathfrak{E}$) such that $y+sum_1^rc_iy_i=f$, and (after a permutation of indices) $1,c_{s+1},...,c_r,0le sle r$ form a
maximal $mathbb{Q}$ linearly independent set of the coefficients $1,c_1,...,c_r$;
(2) there are $nneq 0$, $n_iinmathbb{Z}$, $ginmathfrak{D}$ such that $z^nPi_{i=1}^s z_i^{n_i}=g$.
Furthermore if $(mathfrak{D},X)=(K(z),d/dz)$, then $f$ and $g$ can be chosen to be in $K$, $r=s$, and $c_i=n_i/n$, $i=1,...,r$."
[Risch 1979] Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math 101 (1979) (4) 743-759
I know the mathematical terms field, differential field, field extension, elementary extension, elementary function, and all the mathematical terms in the theorem.
But how can the theorem help in detail to decide if a given $theta$ is algebraic over $mathfrak{D}_1$?
abstract-algebra differential-algebra
asked Jan 27 at 15:51
IV_IV_
1,556525
$endgroup$
add a comment |
$begingroup$
Could you please explain the application of Risch's structure theorem for elementary functions and give some detailed examples?
The Structure Theorem for Elementary Functions:
"Let $(mathfrak{E},Y)$ be elementary over $(mathfrak{D},X)$. kernel $Y= K$. $mathfrak{D}_1=mathfrak{D}(theta_1,...,theta_m)$, $mathfrak{E}=mathfrak{D}_1(theta)$. Let $z_1=e^{y_1},...,z_q=e^{y_q},log z_{q+1}=y_{q+1},...,log z_r=y_r$ be the exponentials and logarithms occurring among $theta_1,...,theta_m$. (See remark following definition at beginning of this part.) Suppose either $theta=z=e^y$ is algebraic over $mathfrak{D}_1$, or $theta=y=log z$ is algebraic over $mathfrak{D}_1$ (where $y$ or respectively $z$ is in $mathfrak{D}_1$). Then
(1) there are $c_iin K$, $fin tilde{mathfrak{D}}$ (= alg. closure of $mathfrak{D}$ in $mathfrak{E}$) such that $y+sum_1^rc_iy_i=f$, and (after a permutation of indices) $1,c_{s+1},...,c_r,0le sle r$ form a
maximal $mathbb{Q}$ linearly independent set of the coefficients $1,c_1,...,c_r$;
(2) there are $nneq 0$, $n_iinmathbb{Z}$, $ginmathfrak{D}$ such that $z^nPi_{i=1}^s z_i^{n_i}=g$.
Furthermore if $(mathfrak{D},X)=(K(z),d/dz)$, then $f$ and $g$ can be chosen to be in $K$, $r=s$, and $c_i=n_i/n$, $i=1,...,r$."
[Risch 1979] Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math 101 (1979) (4) 743-759
I know the mathematical terms field, differential field, field extension, elementary extension, elementary function, and all the mathematical terms in the theorem.
But how can the theorem help in detail to decide if a given $theta$ is algebraic over $mathfrak{D}_1$?
abstract-algebra differential-algebra
asked Jan 27 at 15:51
IV_IV_
1,556525
$endgroup$
Could you please explain the application of Risch's structure theorem for elementary functions and give some detailed examples?
The Structure Theorem for Elementary Functions:
"Let $(mathfrak{E},Y)$ be elementary over $(mathfrak{D},X)$. kernel $Y= K$. $mathfrak{D}_1=mathfrak{D}(theta_1,...,theta_m)$, $mathfrak{E}=mathfrak{D}_1(theta)$. Let $z_1=e^{y_1},...,z_q=e^{y_q},log z_{q+1}=y_{q+1},...,log z_r=y_r$ be the exponentials and logarithms occurring among $theta_1,...,theta_m$. (See remark following definition at beginning of this part.) Suppose either $theta=z=e^y$ is algebraic over $mathfrak{D}_1$, or $theta=y=log z$ is algebraic over $mathfrak{D}_1$ (where $y$ or respectively $z$ is in $mathfrak{D}_1$). Then
(1) there are $c_iin K$, $fin tilde{mathfrak{D}}$ (= alg. closure of $mathfrak{D}$ in $mathfrak{E}$) such that $y+sum_1^rc_iy_i=f$, and (after a permutation of indices) $1,c_{s+1},...,c_r,0le sle r$ form a
maximal $mathbb{Q}$ linearly independent set of the coefficients $1,c_1,...,c_r$;
(2) there are $nneq 0$, $n_iinmathbb{Z}$, $ginmathfrak{D}$ such that $z^nPi_{i=1}^s z_i^{n_i}=g$.
Furthermore if $(mathfrak{D},X)=(K(z),d/dz)$, then $f$ and $g$ can be chosen to be in $K$, $r=s$, and $c_i=n_i/n$, $i=1,...,r$."
[Risch 1979] Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math 101 (1979) (4) 743-759
I know the mathematical terms field, differential field, field extension, elementary extension, elementary function, and all the mathematical terms in the theorem.
But how can the theorem help in detail to decide if a given $theta$ is algebraic over $mathfrak{D}_1$?
abstract-algebra differential-algebra
abstract-algebra differential-algebra
asked Jan 27 at 15:51
IV_IV_
1,556525
asked Jan 27 at 15:51
IV_IV_
1,556525
edited Jan 27 at 21:08
IV_
asked Jan 27 at 15:51
IV_IV_
1,556525
asked Jan 27 at 15:51
IV_IV_
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asked Jan 27 at 15:51
IV_IV_
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1,556525
add a comment |
add a comment |
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