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1
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1answer
210 views

Approximation of a given function by rational functions

Given a function $1/\sqrt{x^2 -k^2}$ where k is a constant with a small imaginary part, how do you go about constructing a rational approximation? I am interested in the L_p (p=2 or $\infty$) norm of ...
12
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2answers
759 views

Determining rational functions by their critical points

Fix an integer $d > 1$ and $2d-2$ points $P_1, \ldots, P_{2d-2}$ in the Riemann sphere (not necessarily distinct). Thanks to the work of Eisenbud and Harris on limit linear series (Inventiones, ...
7
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1answer
953 views

Integer values of a rational function

Suppose we are given a rational function with numerator and denominator being polynomials with integer coefficients. Is there an efficient algorithm for finding all integers arguments at which the ...
4
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1answer
466 views

Why is the inverse of a bijective rational map rational?

Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map ...
6
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1answer
330 views

Asymptotics for the coefficients of a rational function

Let $(a_n)$ be a sequence of non-negative real numbers and assume that the resulting power series defines a rational function $$\sum_{n=0}^\infty a_n x^n = \dfrac{f(x)}{(1-x^{k_1})\cdots ...
6
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3answers
1k views

Rational exponential expressions

Consider the following extension of polynomials. The rational exponential expressions (REXes) are given by: The leaves 1 and $x$ for $x$ drawn from a class of variables; and Closed under the binary ...
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0answers
76 views

Big polynomial similar to a small rational function on a subset of points

Consider a real polynomial $p(x_1,x_2,\dots,x_n)$ that when evaluated on $x_i\in \{0,1\}$ takes values only in $\{0,1\}$. It is clear that $p(x_1,x_2,\dots,x_n)$ can be multilinear (multiaffine) ...