# Questions tagged [rational-functions]

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### Differentiability along hyperplanes for rational functions

This is a follow up to my previous question. Let $f\colon \mathbb R^3\to \mathbb R$ be a continuous function that is rational and differentiable along all planes through $0$, that is, we assume: ...
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### In the rational numbers, is every convergent power series a Taylor series for a rational function?

David Roberts wrote in the comment section of the blog post "Convergence of an infinite sum in the rationals" the following paragraph: Someone mentioned (I think on Twitter) that the Taylor ...
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### When is the product of two elements in algebraic closures of rational functions a constant function?

I have one question on some interactions between sum and product of elements in algebraic clsoures of rational polynomials over algebraically closed fields. My question is as follows: Let E and F be ...
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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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### 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 $p^n$ ...
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### 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 ...
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