The rational-functions tag has no wiki summary.

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### Composition of rational functions

Given a rational function $R\in\Bbb R(x_1,\dots,x_n)$ with multilinear numerator and denominator, is there always a rational function $G\in\Bbb R(x)$ such that $G\circ R\in\Bbb R[x_1,\dots,x_n]$, ...

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### compute the limit of a rational function

Suppose I have a rational function defined by ($s$ complex)
$$
f(s)=w^T s(sI-Q)^{-1} v
$$
for nonzero column vectors $w,v$ and a (large) square matrix $Q$. Further assume that $Q$ is singular and that ...

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### A question on partial fraction decompositions

This question concerns the mapping from the poles of a rational function to the coefficients of its partial fraction decomposition. In general, this mapping is not injective. I want to identify some ...

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### When does the sequence of iterates of a rational function converge?

Darsh asks at the 20-questions seminar:
Let $f:P^1 \rightarrow P^1$ be rational function.
Can you say when the sequence $\{ f^n(x)\}_n=\{ x,f(x),f(f(x)),\cdots\} $ converges? What about the sequence ...

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### A specific polynomial triplet question

Notation
$P_k[n]=\{$multilinear polynomials in $\Bbb R[x_1,x_2,\dots,x_{n-1},x_n]$ of total degree exactly $k\}$.
$k=1$ is just linear polynomials.
QUESTION
Is there a triplet $(p,f,g)\in ...

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### Dimension of certain polynomial spaces

Let $(\omega_1, \eta_1） \dots (\omega_n, \eta_n)$ be $n$ pairs of complex numbers where $\omega_i \ne \omega_j$ for all $1 \leq i \ne j \leq n$. We define the following polynomial space
$$
Z_n^d(\eta, ...

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### Rational dynamical system with nonnegative paramaters

let $A$ be a rational system of the form :$\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ ...

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### When does a rational function have infinitely many integer values for integer inputs?

Consider rational functions $F(x)=P(x)/Q(x)$ with $P(x),Q(x) \in \mathbb{Z}[x]$. I'd like to know when I can expect $F(k) \in \mathbb{Z}$ for infinitely many positive integers $k$. Of course this ...

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### Why is the inverse of a bijective rational map rational?

Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map ...

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### Minimum degree rational function interpolation

Find a rational function $R(x)$ such that:
$1)$ For $i\in\{1,\dots,g\}$, $x_{i}=x_{i-1}+g$ with $x_0=0$.
$2)$ For $i\in\{0,\dots,g-1\}$ $R(x_i)=R(x_i+1)=\dots=R(x_i+g-1)=i+1$.
$3)$ $R(x_g)=g+1$.
...

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### Rational functions and polynomials evaluated on a set of points

Let $S$ be a collection of points on the real line.
Let $\{x_i\}_{i=1}^n$ take values in $S$.
Consider a polynomial $p(x_1,x_2,\dots,x_n)$ over $\mathbb R[x_1,x_2,\dots,x_n]$ of degree $d$ which ...

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### When a sequence of coefficients converges to the coefficients of a rational function $R$, when does the sequence $R_n$ converge uniformly to $R$?

Let $R$ be a rational function of degree $d$ mapping the Riemann sphere to itself:$$R(z) = \frac{a_d z^d + a_{d-1} z^{d-1} + \cdots + a_0}{b_d z^d + b_{d-1} z^{d-1} + \cdots + b_0}$$
where $a_d$ and ...

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### When is a Newton basin fractal continuously determined by the roots of its polynomial?

Newton basin fractals are visualizations of the Julia sets of functions of the form:
$$f_p(z) = z - p(z)/p'(z)$$
where $p$ is a complex polynomial. My question is:
When is the Julia set, ...

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### Truncation error in Padè approximants

Suppose only the following data are known about a rational function $R(x)=P(x)/Q(x)$ (for $P,Q$ polynomials):
(a) the degree of $P$ is $\leq m$ and the degree of $Q$ is $\leq n$;
(b) the first $k$ ...

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### Variation of the argument of a rational function along a circle

I posted this question on MSE a few time ago, but it did not receive much attention. I thought there might be an elementary answer so didn't want to post it directly on MO. My apologies if this ...

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### Markov-type functions

I'd like to have some informations about Markov-type functions (or Cauchy-type):
\[ f(z)=\int_{\Gamma} \frac{\mathrm{d}\gamma(\xi)}{\xi-z}.\]
$\gamma$ is a positive measure with compact support ...

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### Looking for methods/results for explicitly bounding iterations of rational functions

In Theorem 2.6.4 of Beardon's book, "Iteration of Rational Functions", he states the values for the first two coefficients of an iterated power series.
That is, suppose that
$$
...

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### Finding a simpler “local” lower bound for a rational function

I have obtained as the expression for some quantity the following gargantuan formula:
$$ \frac{k^8 + 3k^7 + 8k^6 + 3k^5 - 16k^4 - 32k^3 + 63k^2 - 34k + 6}{k^6 + 3k^5 + 6k^4 - 24k^2 + 21k - 5}$$.
...

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### Approximation of a given function by rational functions

Given a function $1/\sqrt{x^2 -k^2}$ where k is a constant with a small imaginary part, how do you go about constructing a rational approximation? I am interested in the L_p (p=2 or $\infty$) norm of ...

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### Rational maps with all critical points fixed

What can be said about rational self-maps of $\mathbb P^1$
for which all critical points are also fixed points ?
If all but one of the fixed points are critical, there is
a characterization in ...

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### certain trigonometric homeomorphisms

Are there any simple characterizations of rational functions $f(x,y)$ with real coefficients such that $\theta\mapsto f(\cos\theta,\sin\theta)$ is a homeomorphism from $\mathbb R\bmod 2\pi$ to ...

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### Rational interpolation: Error bounds for coefficients

The following question was asked on MSE, but might be more suitable here.
Assume there is a rational function
$$
f:x\mapsto \frac{\sum_{i=0}^m{a_ix^i}}{1+\sum_{j=1}^n{b_jx^j}}
$$
of type $(m,n)$ with ...

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### an equation in fractions

I have an equation of the form $\sum_{i=1}^{m}{\frac{1}{a_{i}-x}}=\sum_{j=1}^{n}{\frac{1}{b_{j}-x}}$ and would like to express $x$ as a an approximate explicit function of the $a_{i},b_{j},m,n$. Have ...

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### Solving a system of rational functions

Given pairwise distinct numbers $c_1, c_2, \dots c_n \in \mathbb{C} \setminus \{0\}$, does the system of equations $$\frac{6}{c_k} + \sum_{i \ne k} \frac{2}{c_k - c_i} = \sum_{i = 1}^n \frac{1}{c_k - ...

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### A curious sequence of rationals: finite or infinite?

Consider the following function repeatedly applied to a rational
$r = a/b$ in lowest terms:
$f(a/b) = (a b) / (a + b - 1)$.
So, $f(2/3) = 6/4 = 3/2$. $f(3/2) = 6/4 = 3/2$.
I am wondering if it is ...

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### Subfields $k\subseteq F\subseteq k(x_1,\dots,x_n)$. Is then $F\cap k[x_1,\dots,x_n]=k(f_1,\dots,f_m)\cap k[x_1,\dots,x_n]$ for polynomials $f_i$?

Let $F\subseteq k(x_1,\dots,x_n)$ be a subfield with $k\subseteq F$. I know that $F=k(\psi_1,\dots,\psi_r)$ for rational functions $\psi_i\in k(x_1,\dots,x_n)$. I'm interested in the intersection ...

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### Approximating rational generating functions

Suppose we have a initial segment $x_1,\ldots,x_N$ (for reasonably large $N$) of a sequence of natural numbers $(x_i)$. We have reason to believe the generating function $\sum_{i=0}^\infty x_iX^i$ is ...

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### Reference request on symmetric polynomials

A version of this question on stackexchange got a few comments from one person and no answers.
Let $e_k$ be the $k$th-degree elementary symmetric polynomial in variables $x_1,\ldots,x_n$ (so in ...

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### sections of vector bundles

Let $X$ a smooth projective connected curve over $\mathbb{C}$.
Let $E$ a vector bundle and $E'$ a subbundle of $E$.
Let $(x_{1},\dots,x_{n})$ $n$ closed points on the curve $X$ with $n>2g$ and $z$ ...

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### Sets of integers represented by degree zero rational functions

Suppose $f(x_1,x_2,\dots)=\frac{P}{Q}$, where $P,Q$ are polynomials in several variables with integer coefficients that have the same degree. Let's denote by $S(f)$ the set of integers $n$ for which ...

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### Properties of a rational function of multiple variables

Suppose you are given a multivariable rational function f(x0,x1,x2,x3,..,xn), so the only four operation are +,-,*,/.
Assume that all constants and exponents are integers within certain range.
I ...

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### Elementary Luroth theorem proof?

Hi, everyone!
I'm trying to explain the proof of Luroth theorem (every field $L$, s.t. $K\subset L\subset K(t)$, is isomorphic to $K(t)$) to the high-school audience. I'm not going to use such ...

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### Is this a Julia set (and if so, for which function family is it the Julia set)?

Consider the function family given by $f_\lambda(z) = z - p_\lambda(z)/p_\lambda'(z)$ where $p_\lambda(z) = (z^2 - 1)(z - \lambda)$. Every attracting cycle and every rational neutral cycle of ...

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### What are good methods for detecting parabolic components and Siegel disk components in the Fatou set of a rational function?

Given a rational function $f$ on the Riemann sphere, I would like to answer the question: Does the Fatou set of the function, $F(f)$, contain any parabolic components or Siegel disk components? ...

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### Generalization of Lagrange inversion with “skewed” formal parameter

I am interested in obtaining an analog of the Lagrange inversion formula, starting from a generalization of the implicit equation. Ordinary Lagrange reversion, as I am familiar with it, starts with ...

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### Integer values of a rational function

Suppose we are given a rational function with numerator and denominator being polynomials with integer coefficients. Is there an efficient algorithm for finding all integers arguments at which the ...

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### Fields over which cubic hypersurfaces are rational

All cubic hypersurfaces having at least one double point are birational to some $P^n$ over an algebraically closed field. How does the statement change as I pass to non alg closed fields? Does it hold ...

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### confusion about rational maps and Fatou components

Dear fellows,
I have come to another conclusion which must be wrong.
Let $f$ be a rational map and let $U$ be a connected but not simply connected open subset of the Fatou set such that $f(U)$ is ...

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### Asymptotics for the coefficients of a rational function

Let $(a_n)$ be a sequence of non-negative real numbers and assume that the resulting power series defines a rational function
$$\sum_{n=0}^\infty a_n x^n = \dfrac{f(x)}{(1-x^{k_1})\cdots ...

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### Determining rational functions by their critical points

Fix an integer $d > 1$ and $2d-2$ points $P_1, \ldots, P_{2d-2}$ in the Riemann sphere (not necessarily distinct). Thanks to the work of Eisenbud and Harris on limit linear series (Inventiones, ...

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### Extrema in two variables of a sum of logs, or equation with sum of rational functions

I am trying to find numerically $\arg\min_{x\in(1, +\infty),y\in(0, 1)}\sum_i\log(xy+\alpha_ix+\beta_iy+\gamma_i)$, where the sum has a large number of terms, and the coefficients are such that the ...

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### Absolute minima for rational function

Suppose I have two rational polynomials $f_1 (x)$ and $f_2 (x)$, both of the form
$$
f_k (x) = \frac{x (A_k x + B_k)}{(C_k x + D_k)}
$$
with constants $A_k, B_k, C_k$ and $D_k$ and where $k=1,2$. I am ...

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### Given a curve, under which condition is the set of gonal morphisms finite

Recently, in my research I bumped onto gonal morphisms. At the moment, my knowledge is based upon some things I read on the internet. Before stating my questions, I added some definitions/facts that ...

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### resolution of singularities and a projection formula

Let $Y$ be a normal surface and let $p:X\longrightarrow Y$ be a resolution of singularities.
Let $f$ be a rational function on $Y$.
Do we have that $p_\ast$div $(d(f\circ p)) = $ div $df$ as ...

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### Can we construct rational functions with prescribed ramification on an algebraic curve over \Qbar

Let $C$ be a smooth projective connected curve of genus $g$ over $\bar{\mathbf{Q}}$. Fix a finite non-empty (Edit) set of closed points $S$ in $C$ and let $U$ be the complement of $S$ in $C$.
Q1. ...

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### Rationality of quadric fibrations

Let $Q\to S$ be a quadric fibration over a rational base $S$, over an algebraically closed field of non zero characteristic. Is it true the following?
$Q$ is rational if and only if $Q \to S$ has a ...

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### Rational exponential expressions

Consider the following extension of polynomials. The rational exponential expressions (REXes) are given by:
The leaves 1 and $x$ for $x$ drawn from a class of variables; and
Closed under the binary ...

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### Singularity structure of integrals of rational functions

Suppose I have a convergent integral of the form $\int_0^1dx_1\dots\int_0^1 dx_n \frac{P(x_i)}{Q(x_i)}$, where $P$ and $Q$ are polynomial functions of $n$ nonnegative real variables $x_i$. Let the ...

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### some rational functions over a field of characteristic 2

I would like to know what are the formal power series $$f(t) = \sum_a \omega_a t^{-a}$$ over an algebraicially closed field of characteristic 2, with two properties: (1) The series represents a ...

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### Pencils with many completely decomposable fibers

Let $F= \frac{G}{H} : \mathbb P^n \to \mathbb P^1$ be a non-constant rational function ($G$ and $H$ homogenous polynomials of the same degree
in $\mathbb C^{n+1})$.
The fiber over $(\lambda:\mu) \in ...