Questions tagged [rational-functions]

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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(...
Carlos Adir's user avatar
2 votes
0 answers
115 views

Embedding $K((X,Y))$ into $K((X))((Y))$

This question is closely related to Explicit elements of $K((x))((y))∖K((x,y))$. Recall that $K((X,Y))$ can be embeded into $K((X))((Y))$, althouth this embedding is not surjective. A natural question ...
Yijun Yuan's user avatar
0 votes
0 answers
38 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 $...
Simón Flavio Ibañez's user avatar
5 votes
1 answer
97 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. ...
D.S. Lipham's user avatar
  • 2,777
1 vote
0 answers
135 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(...
Sam Gue's user avatar
  • 11
2 votes
0 answers
73 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 $...
Troshkin Michael's user avatar
11 votes
1 answer
858 views

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 ...
Madeleine Birchfield's user avatar
1 vote
0 answers
86 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 ...
Luke's user avatar
  • 11
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 ...
T. Amdeberhan's user avatar
8 votes
1 answer
535 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 ...
Georges Elencwajg's user avatar
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0 answers
110 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(...
xiuhua's user avatar
  • 101
9 votes
2 answers
606 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 ...
Ivan Meir's user avatar
  • 4,590
3 votes
0 answers
70 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 ...
Ivan Meir's user avatar
  • 4,590
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 ...
Ivan Meir's user avatar
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11 votes
0 answers
470 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 ...
KhashF's user avatar
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3 votes
1 answer
156 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 ...
KhashF's user avatar
  • 2,227
0 votes
0 answers
57 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,...
Joe's user avatar
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26 votes
3 answers
704 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 $1+b^3$ ...
darij grinberg's user avatar
3 votes
0 answers
40 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 $\...
Per Alexandersson's user avatar
1 vote
1 answer
575 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 ...
user avatar
8 votes
3 answers
859 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 \...
Alice's user avatar
  • 91
5 votes
2 answers
314 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 $ ...
Catologist_who_flies_on_Monday's user avatar
-1 votes
1 answer
86 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\...
ABIM's user avatar
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1 vote
0 answers
43 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 ...
QC_QAOA's user avatar
  • 121
4 votes
2 answers
390 views

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 ...
GA316's user avatar
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1 vote
0 answers
91 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-...
IV_'s user avatar
  • 1,063
6 votes
1 answer
268 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) ...
JSch's user avatar
  • 63
7 votes
1 answer
287 views

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 ...
darij grinberg's user avatar
1 vote
1 answer
116 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 ...
Daps's user avatar
  • 540
1 vote
2 answers
292 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 ...
VoB's user avatar
  • 111
0 votes
1 answer
506 views

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 ...
user avatar
9 votes
0 answers
611 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)\...
user avatar
4 votes
4 answers
504 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(...
zeraoulia rafik's user avatar
0 votes
0 answers
94 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 ...
giulio bullsaver's user avatar
4 votes
1 answer
327 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 ...
mick's user avatar
  • 677
13 votes
1 answer
702 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 ...
Mike Battaglia's user avatar
1 vote
0 answers
319 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\...
Hans's user avatar
  • 2,761
4 votes
1 answer
239 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 $p^n$ ...
P. Grabowski's user avatar
8 votes
1 answer
409 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 ...
Bib-lost's user avatar
  • 267
0 votes
0 answers
99 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},....
Gabriel's user avatar
  • 29
3 votes
2 answers
539 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 ...
Paul Ivanov's user avatar
0 votes
0 answers
43 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 ...
Séverin's user avatar
4 votes
0 answers
169 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,...
Zhi-Wei Sun's user avatar
  • 13.9k
3 votes
1 answer
291 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 ...
Mehdi Yazdi's user avatar
3 votes
1 answer
470 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", ...
user44191's user avatar
  • 4,881
12 votes
1 answer
634 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}...
Ben Webster's user avatar
  • 43.2k
9 votes
0 answers
332 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 $F(t)$ ...
Kaban-5's user avatar
  • 543
1 vote
1 answer
201 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 ...
Loïc Teyssier's user avatar
5 votes
3 answers
292 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 ...
Igor Rivin's user avatar
4 votes
2 answers
3k 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 ...
mathstudent42's user avatar