# Questions tagged [rational-functions]

Rational functions are ratios of two polynomials, for example $(x+5)/(x^2+3)$.

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### Why is $F(x)\cong F[x]\otimes F(x^n)$?

Let $F$ be field. Let $F[x]$ denote the polynomial ring over $F$, and $F(x)$ its field of fractions. Let $F(x^n)$ be the field of fractions of $F[x^n]$, the subring of polynomials in $x^n$. Consider ...
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### Confusion about a rational function

There's a function $f(x)=\frac{(x+2)(x+1)}{(x-4)^2(x+1)}$ I'm a little confused: is this function considered to be defined at $x=-1$? On the one hand, if you plug in $-1$, you'll get $0/0$. But on ...
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### How to find exact value of integral $\int_{0}^{\infty} \frac{1}{\left(x^{4}-x^{2}+1\right)^{n}}dx$?

When I first encountered the integral $\displaystyle \int_{0}^{\infty} \frac{1}{x^{4}-x^{2}+1} d x$, I am very reluctant to solve it by partial fractions and search for any easier methods. Then I ...
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### How to prove or disprove $|r'(a)| \leq c_1 \sup_{x \in [a-1/2,a+1/2]} |r(x)|$?

Let matrix $A = \begin{bmatrix} a & 1 \\ 0 & a \end{bmatrix}$ is a Jordan matrix with $-1 < a < 1$. Let $r(z) = \frac{p(z)}{q(z)}$ is any an irreducible rational function, where ...
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### Does $t^{p-1}$ have an antiderivative in $\Bbb F_q(t)$?

The formal derivative on $\mathbb{F}_p[t]$ is not onto, since $t^{p-1}$ (for instance) has no antiderivative. Does it have one if we extend the formal derivative to $\mathbb{F}_p(t)$? By the quotient ...
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### Is there a general method to operate the reduction of a rational expression to a sum : $\frac {1+2x}{1-3x} \rightarrow 1+5x+\frac{15x^3}{1-3x}$

Source : page $146$ https://archive.org/details/academicalgebraf00bowsrich/page/n5/mode/2up The question I am asking does not deal with partial fraction decomposition ( as far as I can tell). In ...
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### Do holes affect the type of function (even, odd or neither)

We can know whether a function is even or odd by substituting using F(-x) but what if the function has a single hole like this f(x) = $\frac{x(x-2)}{x-2}$ is such a function considered odd or neither? ...
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### Understanding the Single-scale equidistribution theorem for abelian polynomial sequences using example

I wish to understand a remark given in the book 'Higher order fourier analysis' by T. Tao. The remark is related to the following proposition: Proposition 1.1.17 (Single-scale equidistribution theorem ...
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### Proving that a rational function is a section of an invertible sheaf

Let $S$ be a surface, $E$ be a curve on $S$, and $H$ be a hyperplane section of $S$. Let $a\in H^0(S,\mathcal O_S(H+(k-1)E))$, and $b\in H^0(S,\mathcal O_S(H+kE))$. Let $U$ be an open subset of $S$ on ...
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### Factoring when multiplying rational functions

I'm curious why multiplying the numerators of two rational functions before factoring them results in an incorrect solution. Suppose we need to find the product of the following expression in lowest ...
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### Embedding of a variety

Let $D$ be a divisor on a variety $X$, and assume that $h^0(X,\mathcal O_X(D))=n+1$. So let $s_0,\dots,s_n$ be a basis of $H^0(X,\mathcal O_X(D))$. I am trying to understand the following statement : &...
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