The rational-functions tag has no wiki summary.

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### compute the limit of a rational function

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 ...

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### Pencils with many completely decomposable fibers

Let $F= \frac{G}{H} : \mathbb P^n \to \mathbb P^1$ be a non-constant rational function ($G$ and $H$ homogenous polynomials of the same degree
in $\mathbb C^{n+1})$.
The fiber over $(\lambda:\mu) \in ...

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### Singularity structure of integrals of rational functions

Suppose I have a convergent integral of the form $\int_0^1dx_1\dots\int_0^1 dx_n \frac{P(x_i)}{Q(x_i)}$, where $P$ and $Q$ are polynomial functions of $n$ nonnegative real variables $x_i$. Let the ...

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### Dimension of certain polynomial spaces

Let $(\omega_1, \eta_1） \dots (\omega_n, \eta_n)$ be $n$ pairs of complex numbers where $\omega_i \ne \omega_j$ for all $1 \leq i \ne j \leq n$. We define the following polynomial space
$$
Z_n^d(\eta, ...

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### A question on partial fraction decompositions

This question concerns the mapping from the poles of a rational function to the coefficients of its partial fraction decomposition. In general, this mapping is not injective. I want to identify some ...

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### confusion about rational maps and Fatou components

Dear fellows,
I have come to another conclusion which must be wrong.
Let $f$ be a rational map and let $U$ be a connected but not simply connected open subset of the Fatou set such that $f(U)$ is ...

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142 views

### Rational interpolation: Error bounds for coefficients

The following question was asked on MSE, but might be more suitable here.
Assume there is a rational function
$$
f:x\mapsto \frac{\sum_{i=0}^m{a_ix^i}}{1+\sum_{j=1}^n{b_jx^j}}
$$
of type $(m,n)$ with ...

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### Subfields $k\subseteq F\subseteq k(x_1,\dots,x_n)$. Is then $F\cap k[x_1,\dots,x_n]=k(f_1,\dots,f_m)\cap k[x_1,\dots,x_n]$ for polynomials $f_i$?

Let $F\subseteq k(x_1,\dots,x_n)$ be a subfield with $k\subseteq F$. I know that $F=k(\psi_1,\dots,\psi_r)$ for rational functions $\psi_i\in k(x_1,\dots,x_n)$. I'm interested in the intersection ...

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### Truncation error in Padè approximants

Suppose only the following data are known about a rational function $R(x)=P(x)/Q(x)$ (for $P,Q$ polynomials):
(a) the degree of $P$ is $\leq m$ and the degree of $Q$ is $\leq n$;
(b) the first $k$ ...

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143 views

### Solving a system of rational functions

Given pairwise distinct numbers $c_1, c_2, \dots c_n \in \mathbb{C} \setminus \{0\}$, does the system of equations $$\frac{6}{c_k} + \sum_{i \ne k} \frac{2}{c_k - c_i} = \sum_{i = 1}^n \frac{1}{c_k - ...

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431 views

### Properties of a rational function of multiple variables

Suppose you are given a multivariable rational function f(x0,x1,x2,x3,..,xn), so the only four operation are +,-,*,/.
Assume that all constants and exponents are integers within certain range.
I ...

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193 views

### Generalization of Lagrange inversion with “skewed” formal parameter

I am interested in obtaining an analog of the Lagrange inversion formula, starting from a generalization of the implicit equation. Ordinary Lagrange reversion, as I am familiar with it, starts with ...

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### Fields over which cubic hypersurfaces are rational

All cubic hypersurfaces having at least one double point are birational to some $P^n$ over an algebraically closed field. How does the statement change as I pass to non alg closed fields? Does it hold ...

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### Extrema in two variables of a sum of logs, or equation with sum of rational functions

I am trying to find numerically $\arg\min_{x\in(1, +\infty),y\in(0, 1)}\sum_i\log(xy+\alpha_ix+\beta_iy+\gamma_i)$, where the sum has a large number of terms, and the coefficients are such that the ...

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### Rational dynamical system with nonnegative paramaters

let $A$ be a rational system of the form :$\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ ...

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### sections of vector bundles

Let $X$ a smooth projective connected curve over $\mathbb{C}$.
Let $E$ a vector bundle and $E'$ a subbundle of $E$.
Let $(x_{1},\dots,x_{n})$ $n$ closed points on the curve $X$ with $n>2g$ and $z$ ...

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306 views

### Absolute minima for rational function

Suppose I have two rational polynomials $f_1 (x)$ and $f_2 (x)$, both of the form
$$
f_k (x) = \frac{x (A_k x + B_k)}{(C_k x + D_k)}
$$
with constants $A_k, B_k, C_k$ and $D_k$ and where $k=1,2$. I am ...