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0
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1answer
59 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
143 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
784 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
vote
0answers
252 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
202 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
72 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
71 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
163 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
210 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
votes
0answers
128 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
134 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
327 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
71 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
97 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
168 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
votes
0answers
169 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
259 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
vote
0answers
375 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
676 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
vote
0answers
180 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
vote
0answers
151 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
votes
4answers
454 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
votes
0answers
340 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
339 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 ...
12
votes
2answers
822 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
vote
0answers
104 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
votes
0answers
299 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
324 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
votes
3answers
690 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
271 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
320 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
vote
1answer
205 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
votes
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
votes
0answers
241 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
883 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 ...
7
votes
1answer
990 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
854 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
1k 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
357 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
votes
0answers
538 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 ...