# Tagged Questions

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### When is there a polynomial transformation? [on hold]

First part: given $$\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}=\frac{P_3(f(x_1,x_2,\dots,x_n))}{P_4(f(x_1,x_2,\dots,x_n))}|\det (J(f(x_1,x_2,\dots,x_n)))|$$ where $P_i$ is polynomial ( that ...
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### First Description of how to Remove Radicals from Equations

Who first described the technique of removing radicals as indicated in the answers to questions Tools for Removing Radicals from Equations and Rewrite sum of radicals equation as polynomial equation ? ...
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### Is $x^{n}-x-1$ irreducible?

Is it true that for every $n \in \mathbb{N}$, $x^{n}-x-1$ is irreducible in $\mathbb{Z}[x]$? The standard irreducibility criteria seem to fail.
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### irreducible polynomials on the polynomial sequence

I suspect this problem is very famous and it must be studied very well. But I searched in Google and I did not find good reference. I will appreciate any answer and reference for any contribution ...
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### Multivariate Bernstein polynomials for approximation of derivatives.

If I have a $C^\infty$ function $f: [0,1]^n \to \mathbb{R}$ then its Bernstein polynomials  B_m(x) = \sum_{k_1,\dots,k_n=0}^m f\left(\frac{k_1}{m}, \dots, \frac{k_m}{m}\right) \prod_{i=1}^n ...
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### Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus

I was exploring some raising and lowering operators related to an infinitesimal generator for fractional integro-derivatives and found an Appell sequence of polynomials, i.e., an infinite sequence of ...
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### Polynomials all of whose roots are rational

I have two questions about the class of integer-coefficient polynomials all of whose roots are rational. I asked this at MSE, but it attracted little interest (perhaps because it is not interesting!) ...
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### Application of polynomials with non-negative coefficients

Question 1: Are there any deeper applications (in any field of mathematics) of polynomials (with possibly more than one variable) over the real numbers whose coefficients are non-negative? So far I ...
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### Giant Rat of Sumatra singularity

I would be grateful for explanations of the issues raised in any of these three questions, or pointers to the relevant literature (now updated with answers): How did a particular singularity come ...
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### The word problem in the ring of polynomials

This question must be well known but I cannot find it in the literature. Question: What is the computational complexity of the word problem in a subring of the ring of polynomials in $n\ge 1$ ...
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### Polynomials are dense in weighted $L^2$ space

Hi, It seems to be a common knowledge that the polynomials $x^n$ are dense in $L^2$ spaces with various probability weights, such as the gamma distribution weight ...
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### What is known about zero-sets of Schur polynomials?

Consider a set S of partitions not containing the empty partition (I would be happy with, say, all the partitions of size less than k, except for the empty one). Let $U_\lambda^{(r)}$ be the ...
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### quartic diagonal as a sum of squares of quadratic forms

I would appreciate if someone can point out to the literature related to characterizing the set of all different ways to write real quartic diagonal $\sum \limits_{k=1}^n x_k^4, x \in \mathbb{R^n}$ as ...
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### Characteristic polynomials for $K$-Bonacci numbers: what's their name?

Fibonacci numbers are defined by the recurrence relation $f_{n+2}=f_{n+1}+f_{n}$ and Tribonacci numbers by $f_{n+3}=f_{n+2}+f_{n+1}+f_{n}$ One can define, in general, K-Bonacci numbers as ...
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### How large (small) can be the measure of a set where a polynomial takes small values ?

A $n$-th degree polynomial has precisely $n$ roots. So it is natural to ask the question how large ( and small) can be the measure of a set where a polynomial takes small values ? This, and other ...
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### Lines of best fit for the zeros and for the critical points of a univariate polynomial

I have a short note recently accepted to American Math. Monthly but it is all too easy and surely is a rediscovery. The result is that the Lines of best fit for the zeros and for the critical points ...
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### Factorizing polynomials of several variables (in a different perespective)

I am looking for factorization of polynomials of several variables in the way outlined below. Consider a second degree polynomial of two variables over the complex numbers. "P(x,y) = Ax^2 + Bxy + ...
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### Is a polynomial group law on $\mathbb{R}^n$ automatically nilpotent?

I was told that a polynomial group law on (all of) $\mathbb{R}^n$ gives automatically a nilpotent (Lie, of course) group. Is it true? Where can I find a proof? A counterexample for open subsets of ...
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### Variants of Eisenstein irreducibility

In his article where he stated what we know as Eisenstein's irreducibility criterion (which actually was first proved by Schönemann, as was Scholz's reciprocity law and Hensel's Lemma), he ...
Kronecker's paper Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten apparently proves the following result that I'd like to reference: Let $f$ be a monic polynomial with integer ...