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6 added 365 characters in body

Let say we have a function field $k(x,y)$ defined by $f(x,y)$ over $k$, with $\sigma \in Aut(k(x,y)/k)$ and. Suppose, I'm not that out of luck, so that either of $\prod \sigma^i(x)$ or $\sum \sigma^i(x)$ (and the same for $y$) don't fall in $k$, let call them $x^\sigma$ and $y^\sigma$. Then I can compute $k(x^\sigma, y^\sigma)$ and it is not hard to see that if $(\deg(x),\deg(y)) = 1$, that would be my fixed field and I can compute it by finding the min poly of one over another.

However, for example in the case of hyperelliptic involution, we have $\sum \sigma^i(y)=0$. Or there are situations that finding such an $x,y$ is not an easy question. For example, in Algebraic Function Fields and Codes of Stichtenoth, Question 6.9, he asks for such an element $t$ in $F_q(x)$ such that $t^{Aut(F_q(x)/F_q)} = t^{PGL(2,q)}$ is not in $F_q$, and I couldn't solve it (So, it's hard for me at least. I can of course use computer algebra for a particular $q$ but this not what the question asks).

So, I was wondering what is a fail-free way of choosing these generators, such that the fixed field algorithm always works (to prevent them from falling into the constant field and have relatively prime degree). If I use all symmetric polynomials of $Order(\sigma)$ variables, is there a guarantee that at least one of them won't let me down?

Or, if is there better, fixed field computation algorithm there, please tell me (the fixed field algorithm for number field doesn't work straight forward because it could be that $k(x) \not \subseteq k(x^\sigma)$ but one can fixed this if they change the underlying rational function field to the latter, under condition that $x^\sigma$ doesn't fall into $k$, which was my problem to begin with).

Long story short, please tell me what is the fixed field algorithm for automorphisms of (global) function field, that normal people use?

Thanks a lot

post scriptum: I ran into this theorem stated in link text with no proof or reference (beside that Dr. Peter Muller suggested it to the authers (whoever he is)),

[knowing that one can embed a group of automorphisms of rational function field into the field] Let $G = {g_1, . . . , g_m} \subseteq K(x)$ be a finite group. Let $P(t) = \prod^{m}_1 (t−g_i) ∈ K(x)[t]$. Then any non–constant coefficient of $P(t)$ generates $F^G$.

Beside the fact that without having the proof it's hard to generalize it to the nonrational case, it also doesn't guarantee that it doesn't happen that all coeffients of $P(t)$ are constant. In any case, I thought it might help the person who's going to help me ;)

postquam post scriptum: I pasted the "fixed field" functions (for number fields) from both Magma and Pari, here:link text. I see that Magma basically is doing the same thing as I guessed, computing lots of symmetric polynomials and adding them to the base field till the relative degree is the size of the subgroup. For PARI, I don't understand what's the significance of "fixedfieldorbits" and "vandermondeinversemod". I thought It might be helpful. They both lack the function to compute the fixed field of a function field.

Follow-up on @paul garrett's proposed solution to Stichtenoth's problem.

If I understood the proposed method to generate the generator of $F_q(x)^{Aut(F_q(x))}$ correctly, following (sage) code should be able to generate it:

kx.<x> = FunctionField(FiniteField(q))
w = GL2q(Matrix([[0,1],[1,0]]))
Ns = [GL2q(Matrix([[1,n],[0,1]])) for n in range(0,q)]

invElm = (x^q - x)^(q-1)
t = invElm;
for n in Ns:
ninvElm = PGLAction(PGL2GL, GL2PGL.Image(n*w), invElm);
t += ninvElm

print t


Unfortunately, the result is always $(x^q-x)^{(q-1)}$ because summing up over n*w is always zero. Unless, I chose the wrong set of automorphism to apply (this is image of identity plus sum of images of n*w for n =0,..,q-1).

There is another part to that question (that wasn't hard to solved) before asking for finding t. It's to find the ramification locus of $F_q(x)^{Aut(F_q(x))}$ and to prove that all places of deg 2 are conjugates. It probably helps.

But, anyway, my question is not the Stichtenoth's question, I just brought-up it as an example that my problem isn't trivial.

5 deleted 1 characters in body

Let say we have a function field $k(x,y)$ defined by $f(x,y)$ over $k$, with $\sigma \in Aut(k(x,y)/k)$ and. Suppose, I'm not that out of luck, so that either of $\prod \sigma^i(x)$ or $\sum \sigma^i(x)$ (and the same for $y$) don't fall in $k$, let call them $x^\sigma$ and $y^\sigma$. Then I can compute $k(x^\sigma, y^\sigma)$ and it is not hard to see that if $(\deg(x),\deg(y)) = 1$, that would be my fixed field and I can compute it by finding the min poly of one over another.

However, for example in the case of hyperelliptic involution, we have $\sum \sigma^i(y)=0$. Or there are situations that finding such an $x,y$ is not an easy question. For example, in Algebraic Function Fields and Codes of Stichtenoth, Question 6.9, he asks for such an element $t$ in $F_q(x)$ such that $t^{Aut(F_q(x)/F_q)} = t^{PGL(2,q)}$ is not in $F_q$, and I couldn't solve it (So, it's hard for me at least. I can of course use computer algebra for a particular $q$ but this not what the question asks).

So, I was wondering what is a fail-free way of choosing these generators, such that the fixed field algorithm always works (to prevent them from falling into the constant field and have relatively prime degree). If I use all symmetric polynomials of $Order(\sigma)$ variables, is there a guarantee that at least one of them won't let me down?

Or, if is there better, fixed field computation algorithm there, please tell me (the fixed field algorithm for number field doesn't work straight forward because it could be that $k(x) \not \subseteq k(x^\sigma)$ but one can fixed this if they change the underlying rational function field to the latter, under condition that $x^\sigma$ doesn't fall into $k$, which was my problem to begin with).

Long story short, please tell me what is the fixed field algorithm for automorphisms of (global) function field, that normal people use?

Thanks a lot

post scriptum: I ran into this theorem stated in link text with no proof or reference (beside that Dr. Peter Muller suggested it to the authers (whoever he is)),

[knowing that one can embed a group of automorphisms of rational function field into the field] Let $G = {g_1, . . . , g_m} \subseteq K(x)$ be a finite group. Let $P(t) = \prod^{m}_1 (t−g_i) ∈ K(x)[t]$. Then any non–constant coefficient of $P(t)$ generates $F^G$.

Beside the fact that without having the proof it's hard to generalize it to the nonrational case, it also doesn't guarantee that it doesn't happen that all coeffients of $P(t)$ are constant. In any case, I thought it might help the person who's going to help me ;)

postquam post scriptum: I pasted the "fixed field" functions (for number fields) from both Magma and Pari, here:link text. I see that Magma basically is doing the same thing as I guessed, computing lots of symmetric polynomials and adding them to the base field till the relative degree is the size of the subgroup. For PARI, I don't understand what's the significance of "fixedfieldorbits" and "vandermondeinversemod". I thought It might be helpful. They both lack the function to compute the fixed field of a function field.

Follows up

Follow-up on @paul garrett's proposed solution to Stichtenoth's problem.

If I understood the proposed method to generate the generator of $F_q(x)^{Aut(F_q(x))}$ correctly, following (sage) code should be able to generate it:

kx.<x> = FunctionField(FiniteField(q))
w = GL2q(Matrix([[0,1],[1,0]]))
Ns = [GL2q(Matrix([[1,n],[0,1]])) for n in range(0,q)]

invElm = (x^q - x)^(q-1)
t = invElm;
for n in Ns:
ninvElm = PGLAction(PGL2GL, GL2PGL.Image(n*w), invElm);
t += ninvElm

print t


Unfortunately, the result is always $(x^q-x)^{(q-1)}$ because summing up over n*w is always zero. Unless, I chose the wrong set of automorphism to apply (this is image of identity plus sum of images of n*w for n =0,..,q-1).

4 method for rational field not working

Let say we have a function field $k(x,y)$ defined by $f(x,y)$ over $k$, with $\sigma \in Aut(k(x,y)/k)$ and. Suppose, I'm not that out of luck, so that either of $\prod \sigma^i(x)$ or $\sum \sigma^i(x)$ (and the same for $y$) don't fall in $k$, let call them $x^\sigma$ and $y^\sigma$. Then I can compute $k(x^\sigma, y^\sigma)$ and it is not hard to see that if $(\deg(x),\deg(y)) = 1$, that would be my fixed field and I can compute it by finding the min poly of one over another.

However, for example in the case of hyperelliptic involution, we have $\sum \sigma^i(y)=0$. Or there are situations that finding such an $x,y$ is not an easy question. For example, in Algebraic Function Fields and Codes of Stichtenoth, Question 6.9, he asks for such an element $t$ in $F_q(x)$ such that $t^{Aut(F_q(x)/F_q)} = t^{PGL(2,q)}$ is not in $F_q$, and I couldn't solve it (So, it's hard for me at least. I can of course use computer algebra for a particular $q$ but this not what the question asks).

So, I was wondering what is a fail-free way of choosing these generators, such that the fixed field algorithm always works (to prevent them from falling into the constant field and have relatively prime degree). If I use all symmetric polynomials of $Order(\sigma)$ variables, is there a guarantee that at least one of them won't let me down?

Or, if is there better, fixed field computation algorithm there, please tell me (the fixed field algorithm for number field doesn't work straight forward because it could be that $k(x) \not \subseteq k(x^\sigma)$ but one can fixed this if they change the underlying rational function field to the latter, under condition that $x^\sigma$ doesn't fall into $k$, which was my problem to begin with).

Long story short, please tell me what is the fixed field algorithm for automorphisms of (global) function field, that normal people use?

Thanks a lot

post scriptum: I ran into this theorem stated in link text with no proof or reference (beside that Dr. Peter Muller suggested it to the authers (whoever he is)),

[knowing that one can embed a group of automorphisms of rational function field into the field] Let $G = {g_1, . . . , g_m} \subseteq K(x)$ be a finite group. Let $P(t) = \prod^{m}_1 (t−g_i) ∈ K(x)[t]$. Then any non–constant coefficient of $P(t)$ generates $F^G$.

Beside the fact that without having the proof it's hard to generalize it to the nonrational case, it also doesn't guarantee that it doesn't happen that all coeffients of $P(t)$ are constant. In any case, I thought it might help the person who's going to help me ;)

postquam post scriptum: I pasted the "fixed field" functions (for number fields) from both Magma and Pari, here:link text. I see that Magma basically is doing the same thing as I guessed, computing lots of symmetric polynomials and adding them to the base field till the relative degree is the size of the subgroup. For PARI, I don't understand what's the significance of "fixedfieldorbits" and "vandermondeinversemod". I thought It might be helpful. They both lack the function to compute the fixed field of a function field.

Follows up on @paul garrett's proposed solution to Stichtenoth's problem.

If I understood the proposed method to generate the generator of $F_q(x)^{Aut(F_q(x))}$ correctly, following (sage) code should be able to generate it:

kx.<x> = FunctionField(FiniteField(q))
w = GL2q(Matrix([[0,1],[1,0]]))
Ns = [GL2q(Matrix([[1,n],[0,1]])) for n in range(0,q)]

invElm = (x^q - x)^(q-1)
t = invElm;
for n in Ns:
ninvElm = PGLAction(PGL2GL, GL2PGL.Image(n*w), invElm);
t += ninvElm

print t


Unfortunately, the result is always $(x^q-x)^{(q-1)}$ because summing up over n*w is always zero. Unless, I chose the wrong set of automorphism to apply (this is image of identity plus sum of images of n*w for n =0,..,q-1).

3 Link to fixed field function fo Magma and PARI