Questions tagged [rational-functions]
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141 questions
0 votes
0 answers
96 views
Rational function $f(\cdot)$ such that $\sum_{n=2}^{\infty} f(n) \left( \zeta(an)-1 \right)^{2} = q \in \mathbb{Q} \setminus \{0 \}$
Background There are various rational functions such that $$\sum_{n=2}^{\infty} f(n) \left( \zeta(an)-1 \right) = q \label{1} \tag{1}$$ for some $a, q \in \mathbb{Q} \setminus \{ 0 \}$ and $f: \mathbb{...
- 5,025
19 votes
0 answers
608 views
When is the series $\sum_{n=1}^\infty \frac1{a n^2 + b n + c}$ rational?
The question was originally posted on Math Stack Exchange, but no answers were received. Even starting a bounty didn’t get any responses Here Let $a,b,c$ be integers such that $a\neq 0$ and $$ a n^2 +...
- 199
2 votes
3 answers
585 views
Are there known explicit closed-form expressions for the Taylor polynomials of $1 / (1-q)^n$?
Let $$ P_{n,d}(q) := \sum_{k=0}^d \binom{n+k-1}{k} q^k $$ denote the Taylor polynomials (of degree $d$) of $\frac{1}{(1-q)^n}$ (truncated binomial series, the coefficients are the multiset ...
- 7,913
14 votes
2 answers
1k views
A form of implicit function theorem over $ \mathbb Q $
$ \def \Q {\mathbb Q} \def \x {\boldsymbol x} \def \y {\boldsymbol y} \def \a {\boldsymbol a} \def \b {\boldsymbol b} $This question is inspired by the post "Is there any theorem like implicit ...
- 488
0 votes
0 answers
90 views
Positivity of coefficients of inverse of a cumulative distribution function defined by a polynomial
Let $n,m$ be positive integers and consider the polynomial $P(x) = 1-(1-x)^nQ(x)$ where $$Q(x) = \sum_{k=0}^m\binom{n+k-1}{k} x^k$$ is a truncation of the expansion of $(1-x)^{-n}$ around 0. $P(x)$ is ...
18 votes
0 answers
537 views
Rational functions over $\mathbb{Q}$ with rational critical points
Given a natural number $n$, is there always a rational function $f(x)=\frac{g(x)}{h(x)}\in \mathbb{Q}(x)$ of degree $n$ (the degree here being defined as the maximum of numerator and denominator ...
- 3,480
0 votes
2 answers
349 views
Can a variety be the graph of a function in more than one way?
Let $V\subset \Bbb R^n$ be an irreducible affine variety of degree $\ge 2$ and $U_V\subseteq V$ a (Euclidean) open subset. Suppose that $U_V$ is the graph of a rational function, that is, there is an ...
- 14.5k
3 votes
0 answers
250 views
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: ...
- 1,060
3 votes
0 answers
91 views
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 ...
- 366
5 votes
0 answers
161 views
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$...
- 3,758
2 votes
1 answer
200 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 \...
- 187
3 votes
0 answers
128 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-...
- 229
5 votes
1 answer
285 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 $$ |h^{(...
- 187
1 vote
0 answers
302 views
Confusion regarding the invariant rational functions
I was reading S. Mukai's- An Introduction to Invariants and Moduli, and I came across Proposition 6.16 on page 195 (see the screenshot below) It says that "every invariant rational function can ...
- 967
1 vote
0 answers
213 views
On counter-examples to Noether's Problem
Noether's Problem was introduced by Emmy Noether in [4]: Let $\mathsf{k}$ be a field and $K=\mathsf{k}(x_1,\ldots,x_n)$ be a purely transcendental extension. Let $G<S_n$ be a group acting by ...
- 3,758
2 votes
1 answer
166 views
Numerical method with rational nodes and weights to compute exact value of definite integral?
Description Let $p(x)$ be a polynomial of degree $n$ and rational coefficients. I'm interested in computing numerically the exact value of the integral $I$, which is also rational $$I = \int_{a}^{b} p(...
- 273
0 votes
0 answers
75 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
176 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. ...
- 3,691
1 vote
0 answers
237 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 $$E\colon y^2+a_1(...
- 21
2 votes
0 answers
118 views
A closed expression for definite integral of a rational function
Suppose $F(x) = P(x)/Q(x)$ is an integrable rational function on $\mathbb R$, that is, $\deg Q \geq \deg P + 2$, and $Q$ has no real roots. Does there exist an expression for the definite integral $...
- 481
11 votes
1 answer
1k views
In the rational numbers, is every convergent power series a Taylor series for a rational function?
David Roberts wrote in the comment section of the blog post "Convergence of an infinite sum in the rationals" the following paragraph: Someone mentioned (I think on Twitter) that the Taylor ...
user483446
1 vote
0 answers
102 views
When is the product of two elements in algebraic closures of rational functions a constant function?
I have one question on some interactions between sum and product of elements in algebraic clsoures of rational polynomials over algebraically closed fields. My question is as follows: Let E and F be ...
- 11
3 votes
3 answers
1k views
Positivity of a one-variable rational function
Let's consider the $1$-variable rational function $$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 ...
- 43.7k
10 votes
1 answer
771 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 ...
- 48.3k
0 votes
0 answers
141 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(...
- 101
10 votes
2 answers
839 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 ...
- 4,972
2 votes
0 answers
122 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 ...
- 4,972
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 ...
- 4,972
11 votes
0 answers
738 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 ...
- 3,659
3 votes
1 answer
198 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 ...
- 3,659
0 votes
0 answers
82 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,...
- 779
26 votes
3 answers
836 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$ ...
- 35.5k
3 votes
0 answers
70 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} $$ where $\alpha \in \mathbb{R}$, and the other constants are in $\...
- 16.1k
1 vote
1 answer
836 views
Algebraic closure of $\mathbb{C}(t)$
Let $\mathbb{C}(t)$ be the field of rational functions $f(t) = \frac{p(t)}{q(t)}$ with $p,q\in\mathbb{C}[t]$. For instance, the function $g(t) = \sqrt{t}$ does not belong to $\mathbb{C}(t)$ but is ...
user114666
8 votes
3 answers
1k views
residue calculation for rational function
A colleague and I are working on a problem and part of it comes down to evaluating the residue of a rational function. In particular, $$ \mathrm{Res} \left( z^{kn-1} \left( az^{m}+1 \right)^{-k}; r \...
- 91
5 votes
2 answers
455 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
111 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: $$ \|x_n\...
- 4,942
1 vote
0 answers
61 views
Characterization of dimension over $\mathbb{Q}$ of infinite sums of rational functions
Let $P(n)=(n+r_1)(n+r_2)...(n+r_k)$ be a polynomial with simple, rational, negative roots (i.e. $r_i>0$) and degree $k\geq 2$ (I stick with negative roots as I don't have to worry about dividing by ...
- 121
4 votes
2 answers
480 views
Roots of polynomials of particular type
How to find the solutions $x $ of the following equation: $$\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 ...
- 1,319
1 vote
0 answers
127 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-...
- 1,063
6 votes
1 answer
303 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) ...
- 63
8 votes
1 answer
424 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 ...
- 35.5k
1 vote
1 answer
147 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 ...
- 550
1 vote
2 answers
389 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 ...
- 111
0 votes
1 answer
614 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 ...
user157361
10 votes
0 answers
763 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)\...
user147204
4 votes
4 answers
588 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
0 answers
121 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 ...
- 1,473
4 votes
1 answer
442 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 ...
- 799
14 votes
1 answer
1k 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 ...
- 5,005