# Questions tagged [rational-functions]

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### 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 ...
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### 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 ...
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 ...
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 ...
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 ...
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### 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 ... 859 views

1 vote
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### 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 ...
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 ...
1 vote
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### 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-...
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) ...
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 ...
1 vote
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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
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### 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 ...
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 ... 611 views

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### 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 ...
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 ...
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### “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
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### 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 ...
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### 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 ...
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,...
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### 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 ...