# Questions tagged [rational-functions]

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### 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 ...
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### 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 ...
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### 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 $1+b^3$ ...
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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 $\... 1 vote 1 answer 421 views ### Algebraic closure of$\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 ... 8 votes 3 answers 745 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, $$\mathrm{Res} \left( z^{kn-1} \left( az^{m}+1 \right)^{-k}; r \... 5 votes 2 answers 286 views ### Approximation of analytic function by a fixed number of monomials This question seems simple but I can't manage to disprove it. Let N\in \mathbb{N}. We know that by its analyticity that this precise linear combination of monomials \sum_{n=0}^K \frac1{n!} x^n ... -1 votes 1 answer 81 views ### Inferring polynomial rate of convergence from polynomial bound Let x_n be a non-negative valued sequence and suppose that the following hold: \lim\limits_{n\to\infty} x_n =0 There exists some polynomial function p of degree at-least 1 such that:$$ \|x_n\... 1 vote 0 answers 38 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 ... 4 votes 2 answers 341 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 vote 0 answers 86 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-... 6 votes 1 answer 254 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) ... 7 votes 1 answer 248 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 ... 1 vote 1 answer 101 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 ... 1 vote 2 answers 265 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 ... 0 votes 1 answer 451 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 ... 8 votes 0 answers 539 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)\... 486 views

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### How to measure how much a rational function/a singularity of variety is complicated?

There are some theorems about various zeta functions which states the rationality of those. For example, when you consider Igusa's zeta function, roughly the generating series of solutions mod $p^n$ ...
### When does $\sqrt{a_1} + \cdots + \sqrt{a_k} \in K$ imply $\sqrt{a_1}, \ldots, \sqrt{a_k} \in K$?
Consider $R = \mathbb{Z}[X_1, \ldots, X_k]$, the polynomial ring in $k$ variables over $\mathbb{Z}$, and let $S = \mathbb{Z}[\sqrt{X_1}, \ldots, \sqrt{X_k}]$. Then $S/R$ is an integral extension of ...