Expert Answers to Algebra 2 Problems

A rational function f in n variables is a ratio of 2 polynomials,
${\displaystyle f(x_1,\dots x_n)=\frac{p(x_1,\dots x_n)}{q(x_1,\dots x_n)}}$
where q is not identically 0. The function is called symmetric if
${\displaystyle f(x_1,\dots x_n)=f(x_{\sigma \left(1\right)},\dots x_{\sigma \left(n\right)})}$
for any permutation ${\displaystyle \sigma }$ of ${\displaystyle \{1,\dots ,n\}}$.
Let F denote the field of rational functions and S denote the subfield of symmetric rational functions. Suppose the coefficients of polynomials are all real numbers.
Show that F=S(h), where ${\displaystyle h=x_1+2x^2+\dots +n{x}_n}$. In other words, show that h generates F as a field extension of S.
Attempt at Solution:
1.Can't seem to get very far with this one. I know that F is a finite extension of S of degree n! and the Galois group of the extension is ${\displaystyle S_n}$.
2.Using h and the 1st symmetric function ${\displaystyle x_1=x_1+x_2+\dots +x_n}$, we see that ${\displaystyle h-s_1=x_2+2x^3+\dots (n-1)x_nϵS\left(h\right)}$.
3.Can't seem to find a good way to use the other symmetric functions ${\displaystyle s_2,\dots ,s_n}$.

"I have encountered a question like this:
Suppose a complex function f is analytic in |z|<1, continuous in ${\displaystyle \left|z\right|\le 1}$, and |f|=1 on |z|=1. Show that f can be extended as a rational function.
According to the isolation property of zero points, f has only finite zero points in ${\displaystyle \left|z\right|\le 1}$. But I don’t know how to show that it’s a rational function. Hope someone could help.

Let X be an integral and Noetherian scheme . We consider an open subset U of X and ${\displaystyle fϵ\mathrm{\Gamma }(U,O_X)}$. Hence we may regard (U,f) as a rational function in X. I would like to prove that if ${\displaystyle pϵX-U}$(and there is no rational function defined at p that is equivalent to (U,f)), there exists an open subset V of X containing p such that we may write the inverse rational function of (U,f) as (V,g) for certain ${\displaystyle gϵ\mathrm{\Gamma }(V,O_X)}$.
The idea behind this is that if a rational function has a pole at a point, then its inverse rational function has a zero, and in particular it is defined at that point.

Let k be a field, ${\displaystyle A=\frac{k[x,y]}{(y^2,xy)}}$. I know that the only associated points of Spec A are [(x,y)] and [(y)]. I want to show that
${\displaystyle \frac{x-2}{(x-1)x}}$
is not a rational function.
Intuitively it makes sense to me that this function is not defined at the origin [(x,y)], hence not a rational function. But I am having trouble arguing it precisely. I would appreciate any comments. Thanks)

I wanted to find a function f(x) that has the following properties:
1.extreme values at x=-1 and x=0 and x=1
2.x -axis is asymptote
No other asymptotes or extrema.
I found the following two by integrating and trial and error:
${\displaystyle f_1\left(x\right)=x^2{e}^{-x^2}}$
${\displaystyle f_2\left(x\right)=2\ln (x^2+1)+\ln (x^4-x^2+1)}$
How can I find one that is actually a rational function?

Hekp me with regarding this rational function:
${\displaystyle f\left(x\right)=\frac{x-3}{-x+1}}$

Suppose that X is a varity over an algebraically close field K. If I am not mistaken, a rational function over X is a function in K(X). And a function on X is regular at a point ${\displaystyle x\in X}$ if there exists ${\displaystyle g,h\in K\left[X\right]}$, and an open subset ${\displaystyle V\subseteq X}$ containing x, such that for all ${\displaystyle y\in V,h\left(y\right)\ne 0}$ and ${\displaystyle f\left(y\right)=\frac{g\left(y\right)}{h\left(y\right)}}$. And f(x) is a regular function if it is regular at every point in X.
I not understand why rational functions need not be regular, and I am wondering if there is any example of rational function being not regular.

I have
${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{(-1)}^{n}2n{x}^{(2n-1)}}$
It turns out that this series is equal to the function ${\displaystyle \frac{-2x}{{(1+x^2)}^2}}$
Is there a general method that would demonstrate this fact beforehand?

Is the section df associated to a rational function f on a curve X a global section of the canonical sheaf ${\displaystyle {\omega }_X}$? I know its zeroes are the ramification points, but does it have poles?

What is the flaw in the answer for proving any rational function on ${\displaystyle P{P}^1}$ is constant?
Let ${\displaystyle \varphi :P{P}^1\to {\mathrm{\forall }}^1}$ be a rational function. Since ${\displaystyle {\mathrm{\forall }}^1\subset P{P}^1}$ we can think ${\displaystyle \varphi :P{P}^1\to P{P}^1}$ as rational function. As every rational map from ${\displaystyle P{P}^1\to P{P}^n}$ is regular, so ${\displaystyle \varphi }$ is regular. We also know that any regular function on ${\displaystyle P{P}^1}$ is constant, so ${\displaystyle \varphi }$ is constant.

Let f be a rational function on a compact connected Riemann surface X. The rational function f induces a holomorphic map ${\displaystyle \bar{f}:X\to R^1\left(C\right)}$.
Let x be a point on the Riemann sphere ${\displaystyle P^1\left(C\right)}$. ow can I check that if b is a branch point of ${\displaystyle \bar{f}}$ by looking at the derivative of f?
How does this work when ${\displaystyle X=P^1\left(C\right)}$?

This is a pair of similar questions.
Here is the first one:Does there exist a rational function P(x) with a,b ${\displaystyle ϵ\mathbb{Q}}$ such that ${\displaystyle {\int }_a^{b}P\left(x\right)dx}$ is equal to ${\displaystyle k\cdot e}$ for some ${\displaystyle kϵ\mathbb{Q}}$ where e is Euler's number? Prove or disprove that such a function exists. If such a function exists find it. Also, this function should be bounded in the y dimension also so there should exist some r such that |P(x)| < r for every x on [a,b].
This is the second:Let A be a region defined by the inequality 0<P(x,y) where P is a rational function and ${\displaystyle x^2+y^2<r}$ for all ${\displaystyle (x,y)ϵA}$ for some finite r. Does such an A exist such that the area of A equals ${\displaystyle k\cdot e}$ where ${\displaystyle kϵ\mathbb{Q}}$ and e is Euler's constant? Prove or disprove that such an A exists. If such an A exist find P.
Inspiration for the questions:A nice way to approximate pi is by picking random points in the region 0<x<1 and 0<y<1 and seeing if ${\displaystyle x^2+y^2<1}$. This is easy to program because all the functions involved are rational. similar things can be done with ${\displaystyle \ln \left(2\right)}$ for example. I would like to do so with e but can't think of the correct shape to use.
The coefficients of the rational functions should be rational numbers. The bounds on a and b in the first question should be rational.

How do you create a rational function that includes the following: crosses x-axis at 4, Touches the x-axis at -3, One vertical asymptote at x=1 and another at x=6, one horizontal asymptote at y= -3?

If f is quadratic function such that f(0)=1 and ${\displaystyle \int \frac{f\left(x\right)}{x^2{(x+1)}^3}dx}$ is a rational function, find the value of f′(0).
I already tried solving this question by using general quadratic equation ${\displaystyle a{x}^2+bx+c}$ and then using partial fraction method but it became very complicated.

let f(x)=3x^2-4x+2 find f(x+h)-f(x)/h

Write each expression using rational exponents in radical form 1.) 64^2/3 2.) 49^3/2 3.) (-243)^4/5 4.) 625^3/4 5.) (-512)^2/3

Determine whether the function represents exponential growth or exponential decay. Identify the percent rate of change.
y=30(0.95)^t

How do you multiply (5n+6)(5n-5)?

How do you multiply 7(-5v-8)?