# Questions tagged [rational-functions]

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

The motiviation of this question is to look if there is such solution in rational number to the identity which montioned here, I have done many attempts using wolfram alpha to find such pairs of ...
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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 ...
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### 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 ...
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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 ...
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### approximation of rational functions

Suppose $\hat{p}/\hat{q}$ and $p/q$ are two rational functions where $p,q,\hat{p},\hat{q}$ are of degree $n$. Suppose they satisfy that $|p(z)/q(z) - \hat{p}(z)/\hat{q}(z)| < \epsilon$ for any $z$ ...
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### Pythagorean number in Artin's theorem on nonnegative rational fractions

Emil Artin's theorem on nonnegative rational fractions says that a rational fraction $Q$ with $n$ variables with real coefficients which is non-negative on $\mathbb R^n$ is a sum of squares of ...
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### Given a curve $C$, does there exist a rational function on $C$ totally ramified at two given points?

Let $C$ be a smooth projective irreducible curve over $\mathbb C$. Let $x$ and $y$ be distinct points of $C$. We say that $f$ is totally ramified at a point $p$ if the ramification index of $p$ ...
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### Standard rational functions from matrices

In linear algebra we get introduced to standard polynomials that are associated to matrices such as characteristic polynomials and determinants. What are some of the standard rational functions that ...
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### Natural topologies for the space of rational functions

I am looking for natural families of Hausdorff topologies (metrics, norms, if possible) for the space of rational functions of a single complex variable of arbitrary, unbounded denominator degree (and,...
Let $\{f_n\}_{n=0}^\infty$ be a sequence of rational functions of the following form: $$f_n(z) = \sum_{m=1}^\infty \frac{C_{m,n}}{z-m}$$ with $C_{m,n} \in \mathbb{Z}$, $C_{1,n} = 1$, and for each $n \... 7answers 3k views ### Is this a rational function? Is $$\sum_{n=1}^{\infty} \frac{z^n}{2^n-1} \in \mathbb{C}(z)\ ?$$ In a slightly different vein, given a sequence of real numbers$\{a_n\}_{n=0}^\infty$, what are some necessary and sufficient ... 1answer 744 views ### Composition of rational functions Given a rational function$R\in\Bbb R(x_1,\dots,x_n)$with multilinear numerator and denominator, is there always a rational function$G\in\Bbb R(x)$such that$G\circ R\in\Bbb R[x_1,\dots,x_n]$,$G(0)...
Suppose I have a rational function defined by ($s$ complex) $$f(s)=w^T s(sI-Q)^{-1} v$$ for nonzero column vectors $w,v$ and a (large) square matrix $Q$. Further assume that $Q$ is singular and that ...