The rational-functions tag has no wiki summary.
0
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2answers
175 views
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 ...
3
votes
0answers
97 views
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
$$
Z_n^d(\eta, ...
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0answers
54 views
Rational dynamical system with nonnegative paramaters
let $A$ be a rational system of the form :$\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ ...
0
votes
1answer
67 views
Minimum degree rational function interpolation
Find a rational function $R(x)$ such that:
$1)$ For $i\in\{1,\dots,g\}$, $x_{i}=x_{i-1}+g$ with $x_0=0$.
$2)$ For $i\in\{0,\dots,g-1\}$ $R(x_i)=R(x_i+1)=\dots=R(x_i+g-1)=i+1$.
$3)$ $R(x_g)=g+1$.
...
2
votes
1answer
150 views
Rational functions and polynomials evaluated on a set of points
Let $S$ be a collection of points on the real line.
Let $\{x_i\}_{i=1}^n$ take values in $S$.
Consider a polynomial $p(x_1,x_2,\dots,x_n)$ over $\mathbb R[x_1,x_2,\dots,x_n]$ of degree $d$ which ...
6
votes
1answer
800 views
Why is the inverse of a bijective rational map rational?
Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map ...
1
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0answers
259 views
A question on partial fraction decompositions
This question concerns a mapping from the poles of a rational function to its partial fraction decomposition coefficients. We assume that the rational function is the inverse of a polynomial of degree ...
5
votes
1answer
235 views
When is a Newton basin fractal continuously determined by the roots of its polynomial?
Newton basin fractals are visualizations of the Julia sets of functions of the form:
$$f_p(z) = z - p(z)/p'(z)$$
where $p$ is a complex polynomial. My question is:
When is the Julia set, ...
1
vote
0answers
34 views
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 $k$ ...
2
votes
1answer
82 views
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 ...
2
votes
2answers
73 views
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 ...
0
votes
1answer
88 views
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
$$
...
2
votes
2answers
167 views
Finding a simpler “local” lower bound for a rational function
I have obtained as the expression for some quantity the following gargantuan formula:
$$ \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}$$.
...
1
vote
1answer
223 views
Approximation of a given function by rational functions
Given a function $1/\sqrt{x^2 -k^2}$ where k is a constant with a small imaginary part, how do you go about constructing a rational approximation? I am interested in the L_p (p=2 or $\infty$) norm of ...
2
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0answers
131 views
Rational interpolation: Error bounds for coefficients
The following question was asked on MSE, but might be more suitable here.
Assume there is a rational function
$$
f:x\mapsto \frac{\sum_{i=0}^m{a_ix^i}}{1+\sum_{j=1}^n{b_jx^j}}
$$
of type $(m,n)$ with ...
1
vote
0answers
140 views
Solving a system of rational functions
Given pairwise distinct numbers $c_1, c_2, \dots c_n \in \mathbb{C} \setminus \{0\}$, does the system of equations $$\frac{6}{c_k} + \sum_{i \ne k} \frac{2}{c_k - c_i} = \sum_{i = 1}^n \frac{1}{c_k - ...
8
votes
1answer
334 views
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 ...
2
votes
0answers
73 views
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 ...
18
votes
3answers
1k views
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 ...
1
vote
1answer
98 views
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 ...
4
votes
2answers
169 views
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 ...
0
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0answers
174 views
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 $z$ ...
10
votes
1answer
261 views
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 ...
1
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0answers
396 views
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 ...
7
votes
2answers
709 views
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 ...
1
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0answers
188 views
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 ...
1
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0answers
153 views
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 ...
7
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4answers
469 views
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? ...
3
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0answers
348 views
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 ...
6
votes
1answer
344 views
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
$$\sum_{n=0}^\infty a_n x^n = \dfrac{f(x)}{(1-x^{k_1})\cdots ...
13
votes
2answers
862 views
Determining rational functions by their critical points
Fix an integer $d > 1$ and $2d-2$ points $P_1, \ldots, P_{2d-2}$ in the Riemann sphere (not necessarily distinct). Thanks to the work of Eisenbud and Harris on limit linear series (Inventiones, ...
3
votes
2answers
258 views
an equation in fractions
I have an equation of the form $\sum_{i=1}^{m}{\frac{1}{a_{i}-x}}=\sum_{j=1}^{n}{\frac{1}{b_{j}-x}}$ and would like to express $x$ as a an approximate explicit function of the $a_{i},b_{j},m,n$. Have ...
1
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0answers
106 views
Extrema in two variables of a sum of logs, or equation with sum of rational functions
I am trying to find numerically $\arg\min_{x\in(1, +\infty),y\in(0, 1)}\sum_i\log(xy+\alpha_ix+\beta_iy+\gamma_i)$, where the sum has a large number of terms, and the coefficients are such that the ...
0
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0answers
302 views
Absolute minima for rational function
Suppose I have two rational polynomials $f_1 (x)$ and $f_2 (x)$, both of the form
$$
f_k (x) = \frac{x (A_k x + B_k)}{(C_k x + D_k)}
$$
with constants $A_k, B_k, C_k$ and $D_k$ and where $k=1,2$. I am ...
8
votes
1answer
331 views
Given a curve, under which condition is the set of gonal morphisms finite
Recently, in my research I bumped onto gonal morphisms. At the moment, my knowledge is based upon some things I read on the internet. Before stating my questions, I added some definitions/facts that ...
3
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3answers
695 views
When a sequence of coefficients converges to the coefficients of a rational function $R$, when does the sequence $R_n$ converge uniformly to $R$?
Let $R$ be a rational function of degree $d$ mapping the Riemann sphere to itself:$$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}$$
where $a_d$ and ...
2
votes
1answer
274 views
resolution of singularities and a projection formula
Let $Y$ be a normal surface and let $p:X\longrightarrow Y$ be a resolution of singularities.
Let $f$ be a rational function on $Y$.
Do we have that $p_\ast$div $(d(f\circ p)) = $ div $df$ as ...
1
vote
1answer
329 views
Can we construct rational functions with prescribed ramification on an algebraic curve over \Qbar
Let $C$ be a smooth projective connected curve of genus $g$ over $\bar{\mathbf{Q}}$. Fix a finite non-empty (Edit) set of closed points $S$ in $C$ and let $U$ be the complement of $S$ in $C$.
Q1. ...
1
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1answer
206 views
Rationality of quadric fibrations
Let $Q\to S$ be a quadric fibration over a rational base $S$, over an algebraically closed field of non zero characteristic. Is it true the following?
$Q$ is rational if and only if $Q \to S$ has a ...
8
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3answers
2k views
Elementary Luroth theorem proof?
Hi, everyone!
I'm trying to explain the proof of Luroth theorem (every field $L$, s.t. $K\subset L\subset K(t)$, is isomorphic to $K(t)$) to the high-school audience. I'm not going to use such ...
6
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0answers
244 views
Singularity structure of integrals of rational functions
Suppose I have a convergent integral of the form $\int_0^1dx_1\dots\int_0^1 dx_n \frac{P(x_i)}{Q(x_i)}$, where $P$ and $Q$ are polynomial functions of $n$ nonnegative real variables $x_i$. Let the ...
2
votes
1answer
254 views
some rational functions over a field of characteristic 2
I would like to know what are the formal power series $$f(t) = \sum_a \omega_a t^{-a}$$ over an algebraicially closed field of characteristic 2, with two properties: (1) The series represents a ...
6
votes
2answers
911 views
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 ...
8
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1answer
1k views
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 ...
6
votes
3answers
876 views
What characterizes rational functions with nonnegative integer Taylor coefficients?
I believe that there is a statement along the following lines (I would, of course, love to be corrected): a formal power series is the Taylor expansion of a rational function if and only if the ...
6
votes
3answers
2k views
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 ...
5
votes
1answer
358 views
Does an inverse polynomial map on the taylor coefficients of a rational function preserve rationality?
Supppose there are integers $a_1,a_2,\dots$ and a polynomial $p$ so that the integers $p(a_1),p(a_2)...$ satisfy some linear recurrence, i.e. $\sum p(a_i)x^i$ is a rational function of $x$. Must ...
12
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0answers
542 views
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 ...
7
votes
5answers
1k views
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 ...
5
votes
5answers
1k views
When does the sequence of iterates of a rational function converge?
Darsh asks at the 20-questions seminar:
Let f:P^1 -> P^1 be rational function.
Can you say when the sequence ...
