Rational functions are ratios of two polynomials, for example $(x+5)/(x^2+3)$.
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1answer
16 views
Uniqueness theorem for Rational Functions
I know that for polynomials $P,Q$, the equation $P(z) \equiv Q(z)$ implies that they are of the same degree and have the same coefficients. Is there an analogous result for rational fucntions? That ...
5
votes
3answers
76 views
Graph of $\quad\frac{x^3-8}{x^2-4}$.
I was using google graphs to find the graph of $$\frac{x^3-8}{x^2-4}$$ and it gave me:
Why is $x=2$ defined as $3$? I know that it is supposed to tend to 3. But where is the asymptote???
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votes
3answers
62 views
When are the sections of the structure sheaf just morphisms to affine space?
Let $X$ be a scheme over a field $K$ and $f\in\mathscr O_X(U)$ for some (say, affine) open $U\subseteq X$. For a $K$-rational point $P$, I can denote by $f(P)$ the image of $f$ under the map
...
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votes
0answers
20 views
$P(X)-UQ(X)$ is irreducible over $k[U]$ where $U = P/Q$
Let's $P$ and $Q$ in $k[X]$ two polynomials with no common factors, and $U = P/Q$.
How can we prove that $P(X) - U Q(X)$ is irreducible over $k[U]$ ?
I've found some things here : minimal polynomial ...
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0answers
52 views
Laurent expansion of rational functions with NOT polynomial for denominator.
I'm in trouble with the Laurent expansion (and the convergence radius) of a set of rational functions such as:
$$
\frac{1}{ \sin(1/z)}\text{ at }z=0\\
\frac{1}{\exp(1/z)-1}\text{ at }z=0
$$
or the ...
2
votes
1answer
25 views
Questions about ratios
At a school dance, each boy danced with exactly three girls and each girl danced with exactly two boys. if 100 boys attended the school dance, how many girls attended?
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1answer
25 views
Zeros of the analytic limit of complex rational function
For $n\in\mathbb{N}$ let $r_n,\ s_n$ be two polynomials of $O(n)$ degrees with real positive coefficients and set $f_n=r_n/s_n$.
Suppose there exists $c>0$ such that
$\bullet$ if $z\in\mathbb{C}$ ...
2
votes
2answers
102 views
Zeros and poles of rational functions on locally Noetherian schemes
Let $X$ be a locally Noetherian scheme and let $f$ be a rational function on $X$ (i.e. the equivalence class of a pair $(U,f)$, where $f \in \mathcal{O}_X(U)$ and $U$ contains the associated points of ...
5
votes
1answer
41 views
Roots of a polynomial and its derivative
All roots of a complex polynomial have positive imaginary part. Prove that all roots of its derivative also have positive imaginary part.
It's not a homework. This issue has been proposed in the ...
0
votes
2answers
48 views
$X^n - t$ is irreducible over $k(t)$
How can I prove (if it's true) that $X^n - t$ is irreducible over $k(t)$, the field of fractions of $k$ ?
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votes
4answers
131 views
What goes wrong in this derivative?
$$ f(x) = \frac{2}{3} x (x^2-1)^{-2/3} $$
and f'(x) is searched.
So, by applying the product rule $ (uv)' = u'v + uv' $ with $ u=(x^2-1)^{-2/3} $ and $ v=\frac{2}{3} x $, so $ u'=-\frac{4}{3} x ...
3
votes
1answer
67 views
If a rational function is real on the unit circle, what does that say about its roots and poles? Clarification
I'm also self studying the Ahlfors Complex Analysis book.
A question asks:
Suppose $R(z)$ is some rational function which is real on the circle |z|=1 in the complex plane. The question asks, how ...
6
votes
0answers
50 views
Families of curves over number fields
Is there known a number field $K$ and a curve $F(x, y) \in K(t)[x, y]$ such that $F(x, y)$ does not have points over the field of rational functions $K(t)$ but for all but finitely many positive ...
2
votes
0answers
67 views
Finite/algebraic extensions of rational functions
I'm looking for results on the subject of finite/algebraic extensions of rational functions, but I only find papers who deal with algebraic geometry. I only know the basis of Galois theory. Could you ...
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votes
2answers
52 views
Help creating a rational function
Create a rational function with vertical asymptotes $x=\pm1$ and oblique asymptote of $y=2x-3$ and a $y$-intercept of $4$.
