Questions tagged [rational-functions]
The rational-functions tag has no usage guidance.
3
votes
1answer
256 views
Detecting if a series represents a rational function
A lemma by Kronecker states that the series $f(z):= \sum_{i=0}^{\infty} c_i z^i$ represents a rational function if and only if for every $m \gg 0$ the determinant of the following matrix is equal to ...
1
vote
0answers
80 views
Relation between coefficients of expansions
Related to Relations between coefficients of expansions of a rational function at 0 and infinity
I commented at the linked question that the question seemed less about what happened "at infinity", ...
10
votes
1answer
452 views
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}...
8
votes
0answers
261 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
1answer
141 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
3answers
245 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 ...
4
votes
2answers
297 views
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
1answer
339 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
1answer
119 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
1answer
275 views
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
1answer
136 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
0answers
72 views
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
1answer
138 views
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 ...
0
votes
1answer
156 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
0answers
126 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
1answer
365 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
0answers
133 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 $f(p_n)\in\mathbb{N}$...
1
vote
3answers
166 views
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
1answer
135 views
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
1answer
80 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{...
5
votes
1answer
160 views
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
0answers
79 views
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
0answers
152 views
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 ...
13
votes
2answers
565 views
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 ...
2
votes
0answers
71 views
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
1answer
597 views
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
$$
\left(\frac{u(x)}{v(x)}, \frac{s_1(x)+...
2
votes
0answers
105 views
approximation of rational functions
Suppose $\hat{p}/\hat{q}$ and $p/q$ are two rational functions where $p,q,\hat{p},\hat{q}$ are of degree $n$. Suppose they satisfy that $|p(z)/q(z) - \hat{p}(z)/\hat{q}(z)| < \epsilon$ for any $z$ ...
3
votes
1answer
161 views
Pythagorean number in Artin's theorem on nonnegative rational fractions
Emil Artin's theorem on nonnegative rational fractions says that a rational fraction $Q$ with $n$ variables with real coefficients which is non-negative on $\mathbb R^n$ is a sum of squares of ...
2
votes
1answer
241 views
Given a curve $C$, does there exist a rational function on $C$ totally ramified at two given points?
Let $C$ be a smooth projective irreducible curve over $\mathbb C$. Let $x$ and $y$ be distinct points of $C$.
We say that $f$ is totally ramified at a point $p$ if the ramification index of $p$ ...
0
votes
1answer
177 views
Standard rational functions from matrices
In linear algebra we get introduced to standard polynomials that are associated to matrices such as characteristic polynomials and determinants.
What are some of the standard rational functions that ...
7
votes
3answers
465 views
Natural topologies for the space of rational functions
I am looking for natural families of Hausdorff topologies (metrics, norms, if possible) for the space of rational functions of a single complex variable of arbitrary, unbounded denominator degree (and,...
0
votes
0answers
49 views
On sequences of rational functions [duplicate]
Let $\{f_n\}_{n=0}^\infty$ be a sequence of rational functions of the following form: $$ f_n(z) = \sum_{m=1}^\infty \frac{C_{m,n}}{z-m}$$ with $C_{m,n} \in \mathbb{Z}$, $C_{1,n} = 1$, and for each $n \...
18
votes
7answers
3k views
Is this a rational function?
Is $$\sum_{n=1}^{\infty} \frac{z^n}{2^n-1} \in \mathbb{C}(z)\ ?$$
In a slightly different vein, given a sequence of real numbers $\{a_n\}_{n=0}^\infty$, what are some necessary and sufficient ...
1
vote
1answer
670 views
Composition of rational functions
Given a rational function $R\in\Bbb R(x_1,\dots,x_n)$ with multilinear numerator and denominator, is there always a rational function $G\in\Bbb R(x)$ such that $G\circ R\in\Bbb R[x_1,\dots,x_n]$, $G(0)...
1
vote
2answers
229 views
compute the limit of a rational function
Suppose I have a rational function defined by ($s$ complex)
$$
f(s)=w^T s(sI-Q)^{-1} v
$$
for nonzero column vectors $w,v$ and a (large) square matrix $Q$. Further assume that $Q$ is singular and that ...
0
votes
2answers
206 views
A specific polynomial triplet question
Notation
$P_k[n]=\{$multilinear polynomials in $\Bbb R[x_1,x_2,\dots,x_{n-1},x_n]$ of total degree exactly $k\}$.
$k=1$ is just linear polynomials.
QUESTION
Is there a triplet $(p,f,g)\in (P_{k}[4]...
3
votes
0answers
111 views
Dimension of certain polynomial spaces
Let $(\omega_1, \eta_1） \dots (\omega_n, \eta_n)$ be $n$ pairs of complex numbers where $\omega_i \ne \omega_j$ for all $1 \leq i \ne j \leq n$. We define the following polynomial space
$$
Z_n^d(\eta, ...
0
votes
0answers
86 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}} \\ z_{n+1}=\frac{\alpha_{3+}+\beta_{3}x_{n}+\sigma_{3}y_{n}+\lambda_{3}z_{...
0
votes
1answer
164 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
173 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
1k 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 $...
3
votes
0answers
356 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
1answer
418 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, $J(f_p)$...
1
vote
0answers
54 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
163 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 ...
3
votes
2answers
115 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 $\...
1
vote
1answer
97 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
$$
f(z)=az+b_{1}z^{r+1}+\...
2
votes
2answers
224 views
Finding a simpler “local” lower bound for a rational function
I have obtained as the expression for some quantity the following gargantuan formula:
$$ \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}$$.
...
1
vote
1answer
322 views
Approximation of a given function by rational functions
Given a function $1/\sqrt{x^2 -k^2}$ where k is a constant with a small imaginary part, how do you go about constructing a rational approximation? I am interested in the L_p (p=2 or $\infty$) norm of ...
2
votes
0answers
220 views
Rational interpolation: Error bounds for coefficients
The following question was asked on MSE, but might be more suitable here.
Assume there is a rational function
$$
f:x\mapsto \frac{\sum_{i=0}^m{a_ix^i}}{1+\sum_{j=1}^n{b_jx^j}}
$$
of type $(m,n)$ with ...
