Questions tagged [rational-functions]
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100
questions
1
vote
1answer
63 views
Determine whether a rational function on the codomain of a surjective morphism is regular
Let $X$ be a smooth affine algebraic variety with a (not necessarily free) action by an algebraic torus $T$. Let $Y$ be the quotient stack $X/T$ and let $p:X\rightarrow Y$ be the quotient map. Suppose ...
1
vote
2answers
224 views
Chebyshev rational approximation of $e^{x}, x >0$: does it exist?
It's well known that the scalar function $e^x$, for $x \in (-\infty,0]$ can be approximated by Chebyshev rational approximation. In practice, one wants to use a partial fraction decomposition form ...
0
votes
1answer
289 views
How I can prove or disprove that $\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}=1$ in rational? [duplicate]
The motiviation of this question is to look if there is such solution in rational number to the identity which montioned here, I have done many attempts using wolfram alpha to find such pairs of ...
7
votes
0answers
407 views
Again, polynomial bijection $f:{\mathbb Q}\times{\mathbb Q}\times{\mathbb Q}\to {\mathbb Q}$
Assume that there is no polynomial $f(x,y)\in{\mathbb Q}[x,y]{}$ such that $f\colon {\mathbb Q}\times{\mathbb Q}\to {\mathbb Q}$ is a bjiection. Does this imply that there is no polynomial $f(x,y,z)\...
4
votes
4answers
436 views
Does there exist a rational polynomial $P(x)\in{\mathbb Q}[x]{}$ such that $P(\zeta(s))=\zeta(P(s))$?
let $P(x)\in{\mathbb Q}[x]{}$ be a rational polynomial with $P(1) >1$ and $\zeta $ be the Riemann zeta function , I want to know if there exist a rational polynomial such that $P(\zeta(s))=\zeta(P(...
0
votes
0answers
89 views
Trasforming a system of rational equations into an equivalent system of polynomial equations
Suppose that a system of rational equations $r_1=0, r_2=0, \dots, r_m=0$ defines a zero dimensional variety $V$.
Is there an algorithm to produce polynomials $p_i$, starting from the rational ...
4
votes
1answer
286 views
About $a_n = \frac{a_{n-1}(a_{n-1} + C)}{a_{n-2}}.$
Let $A,B,C > 0$. Put $a_1 = A$ and $a_2 = B$ and, for integer $n > 2$,
$$a_n = \frac{a_{n-1}(a_{n-1} + C)}{a_{n-2}}$$
and
$$ T = \lim_{k \to \infty} \frac{a_k}{ a_{k - 1}}.$$
Notice the limit ...
8
votes
0answers
179 views
“Taylor series” is to “Volterra series” as “Padé approximant” is to _________?
Padé approximants are often better than Taylor series at representing a function. Given a Taylor series, one can use Wynn's epsilon algorithm to easily produce the Padé approximants to it.
Volterra ...
1
vote
0answers
130 views
Extend any morphism to suitable projective variety? [closed]
Let $F: X\to \mathbb{P}^n$ be a morphism from an affine variety to projective space (over some algebraically closed field of characteristic zero). Can we always find an open immersion $\iota: X\...
4
votes
1answer
213 views
How to measure how much a rational function/a singularity of variety is complicated?
There are some theorems about various zeta functions which states the rationality of those.
For example, when you consider Igusa's zeta function, roughly the generating series of solutions mod $p^n$ ...
8
votes
1answer
397 views
When does $\sqrt{a_1} + \cdots + \sqrt{a_k} \in K$ imply $\sqrt{a_1}, \ldots, \sqrt{a_k} \in K$?
Consider $R = \mathbb{Z}[X_1, \ldots, X_k]$, the polynomial ring in $k$ variables over $\mathbb{Z}$, and let $S = \mathbb{Z}[\sqrt{X_1}, \ldots, \sqrt{X_k}]$. Then $S/R$ is an integral extension of ...
0
votes
0answers
51 views
Root of the expectation of a random rational function
I am trying to figure out a formula for the unique $\lambda>1$ such that
$$
\mathbb{E}\bigg[\frac{X}{\lambda -X}\bigg]=1
$$
where $X$ is a discrete random variable taking values in $\{\frac{1}{n},....
3
votes
2answers
250 views
Transformation of a fourth degree polynomial
Given $$ P (x) = ax ^ 4 + bx ^ 3 + cx ^ 2 + dx + e = a (x ^ 2 + p_1x + q_1) (x ^ 2 + p_2x + q_2) $$ for some $ a, b, c, d, e, p_1, q_1, p_2, q_2 \in \mathbb R$, prove that $ P (x) $ can be reduced to ...
0
votes
0answers
35 views
Algorithm to determine if a rational fraction has only non negative coefficients
Is there an algorithm that takes as input a polynomial in two variables $P \in \mathbb{N}[x,y]$ and outputs YES if and only if the coefficients of the series $\frac{1}{1-(x+y)} - \frac{1}{1-P}$ are ...
4
votes
0answers
122 views
On sums of minima and maxima
Let $h_1,\ldots,h_n$ be positive integers, and define
$$m(h_1,\ldots,h_n)=\sum_{r_1=0}^{h_1-1}\ldots\sum_{r_n=0}^{h_n-1}\min\left\{\frac{r_1}{h_1},\ldots,\frac{r_n}{h_n}\right\}$$
and
$$M(h_1,\ldots,...
3
votes
1answer
269 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 ...
3
votes
1answer
236 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
517 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}...
9
votes
0answers
304 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
158 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
253 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
1k 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
350 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
126 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
284 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
227 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
77 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
188 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
177 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
414 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
144 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}$...
2
votes
3answers
232 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
249 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)|},$$
...
4
votes
1answer
84 views
Convexity of a set related to certain class of Laurent polynomials
For $r,s\in\mathbb{N}$, let
$$L(z):=\sum_{j=-r}^{s}a_{j}z^{j}$$
be a Laurent polynomial with real coefficients such that there exists a closed curve $\gamma$ encircling the origin, i.e., $0\in\mbox{...
6
votes
1answer
183 views
Surprisingly simple minimum of a rational function on $\mathbb R_+^n$
Motivation:
The following problem has occurred in a study of energy dissipation in a chain of coupled, damped oscillators.
The problem:
Let me define specific rational functions $f$, $g$, and $...
2
votes
0answers
91 views
Galois invariant rational functions and base change
Let $X$ be an algebraic variety defined over a perfect field $k$, let $\bar k$ be
the algebraic closure of $k$, let $X_{\bar k}$ be the base change of $X$ over $\bar k$ and let $G ={\rm Gal}(\bar k/k)$...
3
votes
0answers
178 views
Characterizing rational functions on $\mathbb{Q}$ in terms of smooth extensions to $\mathbb{R}$ and $\mathbb{Q}_p$
Consider a function $f$ from a cofinite subset of $\mathbb{Q}$ to $\mathbb{Q}$. As established here and here $f$ extending to a smooth function on a cofinite subset of $\mathbb{R}$ is not sufficient ...
13
votes
2answers
675 views
When are growth series rational?
For a finitely generated group $G$ with generating set $S$, one can define the word metric. Using this, we are able to define the sequence $\sigma(n)$ by
$ \sigma(n) = |\mathcal{S}(e,n)| $, which is ...
3
votes
0answers
72 views
Nonlinear smooth bijection from $\mathbb Q$ to itself [duplicate]
Is there a bijection $\phi: \mathbb Q \to \mathbb Q$ such that
$\phi$ is nonlinear: different from $ax+b$,
$\phi$ is smooth: the extension $\hat{\phi}$ of $\phi$ over $\mathbb R$ is $\mathcal C^2$ ?
...
4
votes
1answer
609 views
What sort of ind-scheme is this?
It apparently follows from work of Velu (MathSciNet) that every isogeny between elliptic curves in (long) Weierstrass form over $k$ can be written in the form
$$
\left(\frac{u(x)}{v(x)}, \frac{s_1(x)+...
2
votes
0answers
107 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
165 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
254 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
202 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
494 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
50 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 \...
19
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
744 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 $G\circ R\in\Bbb R[x_1,\dots,x_n]$, $G(0)...
0
votes
2answers
229 views
compute the limit of a rational function
Suppose I have a rational function defined by ($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 ...
