Questions tagged [rational-functions]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
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
  • 4,762
11 votes
1 answer
2k views

Integer values of a rational function

Suppose we are given a rational function with numerator and denominator being polynomials with integer coefficients. Is there an efficient algorithm for finding all integers arguments at which the ...
Max Alekseyev's user avatar
15 votes
3 answers
4k views

Elementary Luroth theorem proof?

Hi, everyone! I'm trying to explain the proof of Luroth theorem (every field $L$, s.t. $K\subset L\subset K(t)$, is isomorphic to $K(t)$) to the high-school audience. I'm not going to use such ...
zroslav's user avatar
  • 1,412
14 votes
2 answers
2k views

Determining rational functions by their critical points

Fix an integer $d > 1$ and $2d-2$ points $P_1, \ldots, P_{2d-2}$ in the Riemann sphere (not necessarily distinct). Thanks to the work of Eisenbud and Harris on limit linear series (Inventiones, ...
Xander Faber's user avatar
  • 1,159
14 votes
5 answers
2k views

Rational maps with all critical points fixed

What can be said about rational self-maps of $\mathbb P^1$ for which all critical points are also fixed points ? If all but one of the fixed points are critical, there is a characterization in http://...
Jorge Vitório Pereira's user avatar
13 votes
1 answer
786 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
12 votes
1 answer
643 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
  • 43.9k
9 votes
1 answer
516 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, $J(f_p)$...
Aaron Golden's user avatar
9 votes
2 answers
621 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
  • 4,762
8 votes
3 answers
2k views

What characterizes rational functions with nonnegative integer Taylor coefficients?

I believe that there is a statement along the following lines (I would, of course, love to be corrected): a formal power series is the Taylor expansion of a rational function if and only if the ...
Theo Johnson-Freyd's user avatar
8 votes
3 answers
2k views

Rational exponential expressions

Consider the following extension of polynomials. The rational exponential expressions (REXes) are given by: The leaves 1 and $x$ for $x$ drawn from a class of variables; and Closed under the binary ...
Charles Stewart's user avatar
7 votes
1 answer
502 views

Asymptotics for the coefficients of a rational function

Let $(a_n)$ be a sequence of non-negative real numbers and assume that the resulting power series defines a rational function $$\sum_{n=0}^\infty a_n x^n = \dfrac{f(x)}{(1-x^{k_1})\cdots (1-x^{k_d})}...
tj_'s user avatar
  • 2,160
6 votes
2 answers
2k views

When does a rational function have infinitely many integer values for integer inputs?

Consider rational functions $F(x)=P(x)/Q(x)$ with $P(x),Q(x) \in \mathbb{Z}[x]$. I'd like to know when I can expect $F(k) \in \mathbb{Z}$ for infinitely many positive integers $k$. Of course this ...
A. Pascal's user avatar
  • 1,309
6 votes
1 answer
2k 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 $...
Mikhail Skopenkov's user avatar
5 votes
1 answer
136 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^{(...
xen's user avatar
  • 187
5 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
4 votes
1 answer
321 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 ...
Mike Shulman's user avatar
  • 64.8k
4 votes
1 answer
402 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 ...
Ludwig's user avatar
  • 2,682
1 vote
0 answers
95 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