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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 $(\lambda:\mu) \in ... 0answers 262 views ### Singularity structure of integrals of rational functions Suppose I have a convergent integral of the form$\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$. Let the ... 0answers 107 views ### Dimension of certain polynomial spaces Let$(\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 $$Z_n^d(\eta, ... 0answers 343 views ### A question on partial fraction decompositions This question concerns the mapping from the poles of a rational function to the coefficients of its partial fraction decomposition. In general, this mapping is not injective. I want to identify some "... 0answers 392 views ### confusion about rational maps and Fatou components Dear fellows, I have come to another conclusion which must be wrong. Let f be a rational map and let U be a connected but not simply connected open subset of the Fatou set such that f(U) is ... 0answers 166 views ### Rational interpolation: Error bounds for coefficients The following question was asked on MSE, but might be more suitable here. Assume there is a rational function$$ f:x\mapsto \frac{\sum_{i=0}^m{a_ix^i}}{1+\sum_{j=1}^n{b_jx^j}} $$of type (m,n) with ... 0answers 86 views ### Subfields k\subseteq F\subseteq k(x_1,\dots,x_n). Is then F\cap k[x_1,\dots,x_n]=k(f_1,\dots,f_m)\cap k[x_1,\dots,x_n] for polynomials f_i? Let F\subseteq k(x_1,\dots,x_n) be a subfield with k\subseteq F. I know that F=k(\psi_1,\dots,\psi_r) for rational functions \psi_i\in k(x_1,\dots,x_n). I'm interested in the intersection F\... 0answers 214 views ### Generalization of Lagrange inversion with “skewed” formal parameter I am interested in obtaining an analog of the Lagrange inversion formula, starting from a generalization of the implicit equation. Ordinary Lagrange reversion, as I am familiar with it, starts with ... 0answers 90 views ### approximation of rational functions Suppose \hat{p}/\hat{q} and p/q are two rational functions where p,q,\hat{p},\hat{q} are of degree n. Suppose they satisfy that |p(z)/q(z) - \hat{p}(z)/\hat{q}(z)| < \epsilon for any z ... 0answers 43 views ### Truncation error in Padè approximants Suppose only the following data are known about a rational function R(x)=P(x)/Q(x) (for P,Q polynomials): (a) the degree of P is \leq m and the degree of Q is \leq n; (b) the first k ... 0answers 155 views ### Solving a system of rational functions Given pairwise distinct numbers c_1, c_2, \dots c_n \in \mathbb{C} \setminus \{0\}, does the system of equations$$\frac{6}{c_k} + \sum_{i \ne k} \frac{2}{c_k - c_i} = \sum_{i = 1}^n \frac{1}{c_k - ... 0answers 596 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 ... 0answers 160 views ### 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 ... 0answers 144 views ### 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 ... 0answers 79 views ### 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}} \\ z_{n+1}=\frac{\alpha_{3+}+\beta_{3}x_{n}+\sigma_{3}y_{n}+\lambda_{3}z_{...
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 $z$ ...
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 ...