Questions tagged [rational-functions]
The rational-functions tag has no usage guidance.
133
questions
12
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1
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Relations between coefficients of expansions of a rational function at 0 and infinity
This question goes in the bucket of "this must be well known, but I don't see it and am not sure where to look it up."
Given two Laurent power series $A(t)=\sum_{k>N}a_kt^k$ and $B(t)=\sum_{k>M}...
9
votes
0
answers
336
views
Is Videla's solution of Hilbert's tenth problem for rational functions over field of characteristic 2 wrong?
The paper in question.
Quick introduction to the problem: suppose that $F$ is a finite field of characteristic 2
(for purposes of this post $F = \mathbb{F}_2$ will suffice) and let $F[t]$ and $F(t)$ ...
1
vote
1
answer
207
views
Generating series of rational$\times \exp($rational$)$
It is known that rational functions $f\in \mathbb C(x)$, $0$ not a pole, are the sum of generating series $\sum_{n\geq 0} a_nx^n$ where $(a_n)_n$ is solution of a linear recurrence with constant ...
5
votes
3
answers
305
views
fixed points of quadratic iteration
Consider the well-known iteration $f:z\to z^2 + c,$ and consider the values of $c$ for which $0$ is a periodic point. Experiment shows that most such values of $c$ (about $480$ out of $512$ for period ...
5
votes
2
answers
3k
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approximating the $|x|$ function
I'm familiar with Newman's rational approximation of the absolute value function via rational functions. Are there other explicit functions that approximate $|x|$ with exponential error? I was under ...
4
votes
1
answer
402
views
On a vanishing integral inner product
Let $G(z)$ be an $n\times m$ rational matrix-valued function of full column rank on the unit circle. Further, let $P(z)$ be an $m\times m$ rational matrix-valued function positive definite on the unit ...
0
votes
1
answer
151
views
Are the inverses of a set of quadratic polynomials linearly independent?
Is a collection of reciprocals of monic reducible quadratic polynomials, that is functions of the form
$$ \{ \left( (x-a_i)(x-b_i) \right)^{-1} \}_{i=1}^{k}, $$
linearly independent over a finite ...
-1
votes
1
answer
303
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A simple matrix multiplication query [closed]
The entries of $\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}a'&b'\\c'&d'\end{bmatrix}=\begin{bmatrix}aa'+bc'&ab'+bd'\\ca'+dc'&cb'+dd'\end{bmatrix}$ are curiously given ...
2
votes
1
answer
407
views
Pade approximation of a rational function
I believe I have a naive or hard question because I couldn't find any results in the Internet yet. Any help is greatly appreciated.
So suppose I have two rational functions $R_1(x)$ and $R_2(x)$, i....
0
votes
0
answers
87
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Coefficients of a special meromorphic function
The problem described below appears elementary, but I can't figure out the answer or find it in the literature. I apologize if I have missed something very basic.
Let me begin with considering a ...
4
votes
1
answer
321
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An analogue of rational functions for Hahn series
For any field $k$, we have both the field $k(t)$ of rational functions (formal quotients of polynomials, i.e. the field of fractions of $k[t]$) and the field $k((t))$ of formal Laurent series (which ...
1
vote
1
answer
193
views
Inverse image of a Jordan curve
If there exists a rational map $R$ from the extended plane $\hat{\mathbb{C}}$ to itself, and a Jordan curve $J$ on the plane, such that $R$ has no critical value on the curve, can we say that the ...
4
votes
0
answers
135
views
The class of all iterated antiderivatives of rational functions
Consider the following property of a function $f$:
There exists a non-negative integer $n$ such that the $n$'th derivative of $f$ is a rational function.
Question 1: Is there a name in the ...
12
votes
1
answer
543
views
Splitting of polynomials over rational function fields
Let $K$ be a number field, and let $P(t,X)$ be a monic polynomial in $X$ with coefficients in $K(t)$.
I would like to understand the set $T$ consisting of those $t_0 \in K$ such that the polynomial $...
1
vote
0
answers
225
views
Rational functions and polynomials with infinitely many integer values
Let $f\in \mathbb{C}(x)$ be a rational function.
Assume that we have an infinite collection $\{p_n\}_{n\in \mathbb{N}}$ of positive integers such that for every $n$ it holds that $f(p_n)\in\mathbb{N}$...
3
votes
3
answers
666
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Interpolation by rational functions reference
I have been hearing a lot about a theory of interpolation using rational function, parallel to that of polynomial interpolation.
I'm looking for a book chapter, or even short lecture notes, that will ...
2
votes
1
answer
361
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An upper bound on Zolotarev numbers
Let $k$ be an integer and disjoint closed sets $E,F\subset\mathbb{R}^2$. Consider the Zolotarev number
$$Z_k(E,F):= \inf_{r\in\mathcal{R}_{k,k}}\frac{\sup_{z\in E}|r(z)|}{\inf_{z\in F}|r(z)|},$$
...
4
votes
1
answer
98
views
Convexity of a set related to certain class of Laurent polynomials
For $r,s\in\mathbb{N}$, let
$$L(z):=\sum_{j=-r}^{s}a_{j}z^{j}$$
be a Laurent polynomial with real coefficients such that there exists a closed curve $\gamma$ encircling the origin, i.e., $0\in\mbox{...
7
votes
1
answer
303
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Surprisingly simple minimum of a rational function on $\mathbb R_+^n$
Motivation:
The following problem has occurred in a study of energy dissipation in a chain of coupled, damped oscillators.
The problem:
Let me define specific rational functions $f$, $g$, and $...
2
votes
0
answers
109
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Galois invariant rational functions and base change
Let $X$ be an algebraic variety defined over a perfect field $k$, let $\bar k$ be
the algebraic closure of $k$, let $X_{\bar k}$ be the base change of $X$ over $\bar k$ and let $G ={\rm Gal}(\bar k/k)$...
3
votes
0
answers
196
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Characterizing rational functions on $\mathbb{Q}$ in terms of smooth extensions to $\mathbb{R}$ and $\mathbb{Q}_p$
Consider a function $f$ from a cofinite subset of $\mathbb{Q}$ to $\mathbb{Q}$. As established here and here $f$ extending to a smooth function on a cofinite subset of $\mathbb{R}$ is not sufficient ...
14
votes
2
answers
1k
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When are growth series rational?
For a finitely generated group $G$ with generating set $S$, one can define the word metric. Using this, we are able to define the sequence $\sigma(n)$ by
$ \sigma(n) = |\mathcal{S}(e,n)| $, which is ...
3
votes
0
answers
90
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Nonlinear smooth bijection from $\mathbb Q$ to itself [duplicate]
Is there a bijection $\phi: \mathbb Q \to \mathbb Q$ such that
$\phi$ is nonlinear: different from $ax+b$,
$\phi$ is smooth: the extension $\hat{\phi}$ of $\phi$ over $\mathbb R$ is $\mathcal C^2$ ?
...
4
votes
1
answer
630
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What sort of ind-scheme is this?
It apparently follows from work of Velu (MathSciNet) that every isogeny between elliptic curves in (long) Weierstrass form over $k$ can be written in the form
$$
\left(\frac{u(x)}{v(x)}, \frac{s_1(x)+...
2
votes
0
answers
113
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approximation of rational functions
Suppose $\hat{p}/\hat{q}$ and $p/q$ are two rational functions where $p,q,\hat{p},\hat{q}$ are of degree $n$. Suppose they satisfy that $|p(z)/q(z) - \hat{p}(z)/\hat{q}(z)| < \epsilon$ for any $z$ ...
3
votes
1
answer
208
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Pythagorean number in Artin's theorem on nonnegative rational fractions
Emil Artin's theorem on nonnegative rational fractions says that a rational fraction $Q$ with $n$ variables with real coefficients which is non-negative on $\mathbb R^n$ is a sum of squares of ...
2
votes
1
answer
294
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Given a curve $C$, does there exist a rational function on $C$ totally ramified at two given points?
Let $C$ be a smooth projective irreducible curve over $\mathbb C$. Let $x$ and $y$ be distinct points of $C$.
We say that $f$ is totally ramified at a point $p$ if the ramification index of $p$ ...
0
votes
1
answer
283
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Standard rational functions from matrices
In linear algebra we get introduced to standard polynomials that are associated to matrices such as characteristic polynomials and determinants.
What are some of the standard rational functions that ...
7
votes
3
answers
593
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Natural topologies for the space of rational functions
I am looking for natural families of Hausdorff topologies (metrics, norms, if possible) for the space of rational functions of a single complex variable of arbitrary, unbounded denominator degree (and,...
0
votes
0
answers
53
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On sequences of rational functions [duplicate]
Let $\{f_n\}_{n=0}^\infty$ be a sequence of rational functions of the following form: $$ f_n(z) = \sum_{m=1}^\infty \frac{C_{m,n}}{z-m}$$ with $C_{m,n} \in \mathbb{Z}$, $C_{1,n} = 1$, and for each $n \...
19
votes
7
answers
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Is this a rational function?
Is $$\sum_{n=1}^{\infty} \frac{z^n}{2^n-1} \in \mathbb{C}(z)\ ?$$
In a slightly different vein, given a sequence of real numbers $\{a_n\}_{n=0}^\infty$, what are some necessary and sufficient ...
1
vote
1
answer
988
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Composition of rational functions
Given a rational function $R\in\Bbb R(x_1,\dots,x_n)$ with multilinear numerator and denominator, is there always a rational function $G\in\Bbb R(x)$ such that $G\circ R\in\Bbb R[x_1,\dots,x_n]$, $G(0)...
0
votes
2
answers
252
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compute the limit of a rational function
Suppose I have a rational function defined by ($s$ complex)
$$
f(s)=w^T s(sI-Q)^{-1} v
$$
for nonzero column vectors $w,v$ and a (large) square matrix $Q$. Further assume that $Q$ is singular and that ...
0
votes
2
answers
254
views
A specific polynomial triplet question
Notation
$P_k[n]=\{$multilinear polynomials in $\Bbb R[x_1,x_2,\dots,x_{n-1},x_n]$ of total degree exactly $k\}$.
$k=1$ is just linear polynomials.
QUESTION
Is there a triplet $(p,f,g)\in (P_{k}[4]...
3
votes
0
answers
116
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Dimension of certain polynomial spaces
Let $(\omega_1, \eta_1) \dots (\omega_n, \eta_n)$ be $n$ pairs of complex numbers where $\omega_i \ne \omega_j$ for all $1 \leq i \ne j \leq n$. We define the following polynomial space
$$
Z_n^d(\eta, ...
0
votes
0
answers
124
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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}} \\ z_{n+1}=\frac{\alpha_{3+}+\beta_{3}x_{n}+\sigma_{3}y_{n}+\lambda_{3}z_{...
0
votes
1
answer
231
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
1
answer
213
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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 ...
6
votes
1
answer
2k
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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 $...
3
votes
0
answers
398
views
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 "...
9
votes
1
answer
516
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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, $J(f_p)$...
1
vote
0
answers
71
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
1
answer
254
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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 ...
4
votes
2
answers
135
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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 $\...
1
vote
1
answer
105
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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
$$
f(z)=az+b_{1}z^{r+1}+\...
3
votes
2
answers
268
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Finding a simpler "local" lower bound for a rational function
I have obtained as the expression for some quantity the following gargantuan formula:
$$ \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}$$.
...
2
votes
2
answers
430
views
Approximation of a given function by rational functions
Given a function $1/\sqrt{x^2 -k^2}$ where k is a constant with a small imaginary part, how do you go about constructing a rational approximation? I am interested in the L_p (p=2 or $\infty$) norm of ...
2
votes
0
answers
287
views
Rational interpolation: Error bounds for coefficients
The following question was asked on MSE, but might be more suitable here.
Assume there is a rational function
$$
f:x\mapsto \frac{\sum_{i=0}^m{a_ix^i}}{1+\sum_{j=1}^n{b_jx^j}}
$$
of type $(m,n)$ with ...
1
vote
0
answers
170
views
Solving a system of rational functions
Given pairwise distinct numbers $c_1, c_2, \dots c_n \in \mathbb{C} \setminus \{0\}$, does the system of equations $$\frac{6}{c_k} + \sum_{i \ne k} \frac{2}{c_k - c_i} = \sum_{i = 1}^n \frac{1}{c_k - ...
8
votes
1
answer
442
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 $\...