# Questions tagged [rational-functions]

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137
questions

3
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### Differentiability along hyperplanes for rational functions

This is a follow up to my previous question.
Let $f\colon \mathbb R^3\to \mathbb R$ be a continuous function that is rational and differentiable along all planes through $0$, that is, we assume:
...

0
votes

0
answers

31
views

### Convergence region of multivariate rational functions

Assume $p, q \in \mathbb{R}[x_1,\ldots,x_k]$ and let $ \vec{0} \not\in V(q) := \{\vec{x} \in \mathbb{R}^k \mid q(\vec{x}) = 0 \}$ such that $r_q := \inf_{\vec{x}\in V(q)} |\!|\vec{x}|\!|_\infty < \...

2
votes

0
answers

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### What circumstances guarantee a p-adic affine conjugacy map will be a rational function?

Let $\Bbb Q_p$ be a p-adic field and let any element $x$ of $\Bbb Q_p$ be associated with a unique element of $\Bbb Z_p$ via the quotient / equivalence relation $\forall n\in\Bbb Z:p^nx\sim x$
Then in ...

5
votes

0
answers

90
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### Noether's Problem and the Inverse Problem on Galois Theory

For the sake of simplicity, assume the base field $k$ as having zero characteristic. I will discuss 4 different formulations of Noether's Problem.
version 1 - original Noether's problem: Let $G<S_n$...

0
votes

0
answers

101
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### A question about the existence of rational functions

I am reading a paper Representations of shifted quantum affine algebras. I have a question about the existence of a rational function about the remark $4.4$
I'll briefly describe the problem.
We let $...

2
votes

1
answer

107
views

### Bound for the $n$-th derivative of a proper rational function with no poles on the right half-plane

Suppose that
$f$ and $g$ are polynomials with nonnegative coefficients,
the degree of $g$ is greater than the degree of $f$,
$g + f$ have no zeros on the right half plane $\mathbb{C}_+ = \{z \in \...

2
votes

0
answers

92
views

### Unirationality connected with $S$-unit equation

This update of question asked before.
Let $n$ be a natural number. Consider a subvariety in $\mathbb A^{3n+2}$ (say over $\mathbb C$) given by the equation
$$x_1(t-y_1)\dots (t-y_n)+x_2(t-z_1)\dots(t-...

5
votes

1
answer

148
views

### Upper bound for the $n$-th derivative of a rational function $\frac{f}{f+g}$

Let $f$ and $g$ be real polynomials with nonnegative coefficients. Let
$$
h = \frac{f}{f+g}.
$$
I want to prove that the $n$-th derivative of $h$ satisfies:
There exists $C > 0$ such that
$$
|h^{(...

1
vote

0
answers

234
views

### Confusion regarding the invariant rational functions

I was reading S. Mukai's- An Introduction to Invariants and Moduli, and I came across Proposition 6.16 on page 195 (see the screenshot below)
It says that "every invariant rational function can ...

1
vote

0
answers

124
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### On counter-examples to Noether's Problem

Noether's Problem was introduced by Emmy Noether in [4]:
Let $\mathsf{k}$ be a field and $K=\mathsf{k}(x_1,\ldots,x_n)$ be a purely transcendental extension. Let $G<S_n$ be a group acting by ...

2
votes

1
answer

97
views

### Numerical method with rational nodes and weights to compute exact value of definite integral?

Description
Let $p(x)$ be a polynomial of degree $n$ and rational coefficients.
I'm interested in computing numerically the exact value of the integral $I$, which is also rational
$$I = \int_{a}^{b} p(...

0
votes

0
answers

45
views

### Educated guess for algebraic approximation

I found a very neat ancient hindi formula for approximating square roots using rational numbers. After doing some algebra on the formula, i came across with this recursive relation:
Given any number $...

5
votes

1
answer

116
views

### Jordan curve boundaries of Fatou components

Let $f:\mathbb C\to \mathbb C$ be a rational map and let $J(f)$ and $F(f)$ denote the Julia and Fatou sets of $f$, respectively.
Let $\mathcal S$ be the set of all boundaries of Fatou components. ...

1
vote

0
answers

162
views

### How to link the rank of Elliptic Surfaces and Elliptic Curves over function fields?

I have been investigating certain elliptic surfaces for my research, and when giving a presentation, I was asked why when given an elliptic surface $E$, say given in Weierstrass form
$$E\colon y^2+a_1(...

2
votes

0
answers

84
views

### A closed expression for definite integral of a rational function

Suppose $F(x) = P(x)/Q(x)$ is an integrable rational function on $\mathbb R$, that is, $\deg Q \geq \deg P + 2$, and $Q$ has no real roots.
Does there exist an expression for the definite integral $...

11
votes

1
answer

1k
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### In the rational numbers, is every convergent power series a Taylor series for a rational function?

David Roberts wrote in the comment section of the blog post "Convergence of an infinite sum in the rationals" the following paragraph:
Someone mentioned (I think on Twitter) that the Taylor ...

1
vote

0
answers

87
views

### When is the product of two elements in algebraic closures of rational functions a constant function?

I have one question on some interactions between sum and product of elements in algebraic clsoures of rational polynomials over algebraically closed fields.
My question is as follows:
Let E and F be ...

3
votes

3
answers

1k
views

### Positivity of a one-variable rational function

Let's consider the $1$-variable rational function
$$F(z):=\frac{1-z}{(z^3 - z^2 + 2z - 1)\,(z^3 + z^2 + z - 1)}.$$
Numerical evidence convinces me of the truth of the following.
QUESTION. Can you ...

8
votes

1
answer

624
views

### What are the rational functions on a noetherian affine scheme?

Let $A$ be a noetherian ring and $X=\operatorname {Spec}A$ the corresponding affine scheme.
There are three rings which might reasonably be called the ring of rational functions on $X$.
a) The total ...

0
votes

0
answers

112
views

### How to prove or disprove $ |r'(a)| \leq c_1 \sup_{x \in [a-1/2,a+1/2]} |r(x)|$?

Let $ A = \begin{bmatrix}
a & 1 \\ 0 & a
\end{bmatrix}$ be a Jordan matrix with $ -1 < a < 1 $.
Let $r(z) = \frac{p(z)}{q(z)}$ be an irreducible rational function, where $p(...

9
votes

2
answers

635
views

### Is $\mathbb{Q}$ the orbit of a rational function under iteration?

In this previous post I asked for the smallest set of continuous real functions that could generate $\mathbb Q$ by iteration starting from $0$. Surprisingly one continuous function suffices.
In the ...

2
votes

0
answers

79
views

### Is there a finite set of polynomials generating all rational numbers by iteration?

In this previous post I asked for the smallest set of continuous real functions that could generate $\mathbb{Q}$ by iteration starting from 0. Surprisingly one continuous function suffices.
The ...

33
votes

2
answers

2k
views

### What is the smallest set of real continuous functions generating all rational numbers by iteration?

I recently came across this problem from USAMO 2005:
"A calculator is broken so that the only keys that still
work are the $\sin$, $\cos$, $\tan$, $\arcsin$, $\arccos$ and $\arctan$ buttons. The ...

11
votes

0
answers

541
views

### A curious observation on the elliptic curve $y^2=x^3+1$

Here is a calculation regarding the $2$-torsion points of the elliptic curve $y^2=x^3+1$ which looks really miraculous to me (the motivation comes at the end).
Take a point of $y^2=x^3+1$ and ...

3
votes

1
answer

164
views

### A question about decompositions of rational functions

Let $f_1,g_1,f_2,g_2$ be non-constant rational functions on the Riemann sphere (i.e. elements of $\Bbb{C}(z)-\Bbb{C}$) satisfying $f_1\circ g_1=f_2\circ g_2$. Suppose there is a prime number $p$ such ...

0
votes

0
answers

58
views

### Arranging the $k$ solutions of $r(z)=te^{i\theta}$ into $k$ continuous functions of $(t,\theta)$

I have originally opened this question on MSE, but I migrated here, since I realized this environment is more suitable.
Let $r$ be a rational function, that is, quotient of two coprime polynomials $p,...

26
votes

3
answers

716
views

### Subtraction-free identities that hold for rings but not for semirings?

Here is a concrete, if seemingly unmotivated, aspect of the question I am interested in:
Question 1. Let $a$ and $b$ be two elements of a (noncommutative) semiring $R$ such that $1+a^3$ and $1+b^3$ ...

3
votes

0
answers

47
views

### Convexity of integral trajectories of rational vector field

Suppose we have a vector field determined by a rational function,
of the form
$$
R(z) = \alpha i z + \sum_{j=1}^k \frac{c_j}{z-r_j}
$$
where $\alpha \in \mathbb{R}$, and the other constants are in $\...

1
vote

1
answer

707
views

### Algebraic closure of $\mathbb{C}(t)$

Let $\mathbb{C}(t)$ be the field of rational functions $f(t) = \frac{p(t)}{q(t)}$ with $p,q\in\mathbb{C}[t]$.
For instance, the function $g(t) = \sqrt{t}$ does not belong to $\mathbb{C}(t)$ but is ...

8
votes

3
answers

967
views

### residue calculation for rational function

A colleague and I are working on a problem and part of it comes down to evaluating the residue of a rational function. In particular,
$$
\mathrm{Res} \left( z^{kn-1} \left( az^{m}+1 \right)^{-k}; r \...

5
votes

2
answers

325
views

### Approximation of analytic function by a fixed number of monomials

This question seems simple but I can't manage to disprove it. Let $N\in \mathbb{N}$. We know that by its analyticity that this precise linear combination of monomials
$
\sum_{n=0}^K \frac1{n!} x^n
$
...

-1
votes

1
answer

87
views

### Inferring polynomial rate of convergence from polynomial bound

Let $x_n$ be a non-negative valued sequence and suppose that the following hold:
$\lim\limits_{n\to\infty} x_n =0$
There exists some polynomial function $p$ of degree at-least $1$ such that:
$$
\|x_n\...

1
vote

0
answers

45
views

### Characterization of dimension over $\mathbb{Q}$ of infinite sums of rational functions

Let $P(n)=(n+r_1)(n+r_2)...(n+r_k)$ be a polynomial with simple, rational, negative roots (i.e. $r_i>0$) and degree $k\geq 2$ (I stick with negative roots as I don't have to worry about dividing by ...

4
votes

2
answers

414
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### Roots of polynomials of particular type

How to find the solutions $x $ of the following equation: $$\frac{n_1}{x + n_1} + \frac{n_2}{x + n_2} + \cdots +\frac{n_k}{x + n_k} = 1$$ where $n_i$s are natural numbers.
For the case $k=2$, I get ...

1
vote

0
answers

98
views

### Extension of closed-form solvability of polynomial equations of x and exp to rational expressions?

Is my conjecture below true?
It's a new conjecture. It's an extension of Lin's theorem in [Lin 1983] which is cited in [Chow 1999] - see both references at the bottom of this page.
It seems that Ferng-...

6
votes

1
answer

274
views

### Algebraic geometry additionally equipped with field automorphism operation

I am looking for some facts on theory, which is essentially algebraic geometry but with field automorphisms added as 'basic' operations. (Precisely, I mean universal algebraic geometry for (universal) ...

7
votes

1
answer

322
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### Constructive definition of noncommutative rational functions (aka free skew fields)

The question
Let $F$ be a field. (I am fine with assuming $F=\mathbb{Q}$, but I suspect
that a "right" answer will be independent of $F$.) Let $k$ be a nonnegative integer.
Question. Is ...

1
vote

1
answer

121
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### 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

2
answers

315
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

1
answer

528
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### How I can prove or disprove that $\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}=1$ has solutions in rationals? [duplicate]

The motivation of this question is to look if there is such solution in rational number to the identity which mentioned here, I have done many attempts using Wolfram Alpha to find such pairs of ...

9
votes

0
answers

657
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

4
answers

508
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### 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

0
answers

97
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

1
answer

349
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 ...

13
votes

1
answer

844
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

0
answers

385
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

1
answer

248
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

1
answer

419
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

0
answers

110
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### 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

2
answers

654
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 ...