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Questions tagged [rational-functions]

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3
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 ...