**234**

votes

**15**answers

32k views

### Why do roots of polynomials tend to have absolute value close to 1?

While playing around with Mathematica I noticed that most polynomials with real coefficients seem to have most complex zeroes very near the unit circle. For instance, if we plot all the roots of a ...

**2**

votes

**1**answer

50 views

### Bound the degree of the generator of polynomial ring

Suppose we are given two polynomial rings $R_1$ and $R_2$ by presenting their generators, $S_1$ and $S_2$, where $S_i$ are finite set of $m_i$ variables, $i.e.$, $S_i\subset ...

**48**

votes

**2**answers

1k views

### Does one real radical root imply they all are?

Is there an example of an irreducible polynomial $f(x) \in \mathbb{Q}[x]$ with a real root expressible in terms of real radicals and another real root not expressible in terms of real radicals?

**5**

votes

**1**answer

204 views

### Modular polynomials for elliptic curves point counting

The Schoof-Elkies-Atkin (SEA) algorithm (for counting points on elliptic curves over a finite field) performs computations over polynomials modulo some modular polynomials. Originally the "classical" ...

**1**

vote

**0**answers

147 views

### On monotonicity of the roots of polynomials

I feel that the question that I am posting is not a general research level question but probably some special form of it where the behaviors of the roots of polynomials are studied.
Here is my ...

**2**

votes

**1**answer

293 views

### Is the identification between symmetric tensors and homogeneous polynomials useful?

The general question:
Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification
$$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$
between the space of symmetric order ...

**1**

vote

**0**answers

101 views

### Puzzling CAS-detected factorization by cyclotomic polynomials [closed]

If one factorizes by CAS the expression $$x^{\frac{m(m+1)}{2}}\prod_{k=1}^m(x^k+(\frac{1}{x})^k)$$ a puzzling perfect factorization seems to be possible for all natural values of $m$.
E.g. for ...

**2**

votes

**1**answer

102 views

### using polynomials as lower / upper bound?

I'm interested in the question of given a differentiable and bounded function $f(\vec{x})$ (over a single variable or multiple variables, over a bounded domain $D$), finding a pair of polynomials ...

**10**

votes

**1**answer

646 views

### Ways to show a system of polynomial equations has no solution

I came across the following system of polynomial equations on $X_1,\dots,X_{m-2}$:
$$
\begin{cases}
2X_{2s}+\sum\limits_{t=1}^{2s-1}(-1)^tX_tX_{2s-t}=0,\quad s=1,\dots,\frac{m}{2}-1,\\
...

**0**

votes

**1**answer

216 views

### Decomposition of symmetric homogeneous polynomials

Can every symmetric polynomial of degree $r$ in $d$ variables that has no constant term be written as a sum of the $r$th powers of linear polynomials in $d$ variables and a homogeneous polynomial of ...

**17**

votes

**2**answers

515 views

### Stability of real polynomials with positive coefficients

Say that a polynomial in an indeterminate $x$ with real coefficients of degree $d$ has positive coefficients if each of the coefficients of $x^d,\ldots,x^1,x^0$ is (strictly) positive.
For $f$ a ...

**2**

votes

**2**answers

400 views

### Looking for ways how to calculate $\Phi_n(i)$

$Φ_n(1)$ and $Φ_n(−1)$ for the cyclotomic polynomials are well-known.
I am now looking for
$$Φ_n(i)$$
and/or
$$Φ_n(−i)$$
with i the complex unit.
At this moments my endeavours result in intricate ...

**1**

vote

**0**answers

196 views

### Solving polynomials of arbitrary degree

Is there any analogue of the method for expressing roots of polynomials of degree 5 with elliptic and η-functions that generalizes to polynomials of degrees n>5?
More specifically: does there exist ...

**4**

votes

**1**answer

354 views

### q-th powers and roots of polynomials

Let $p,q,r$ be integers with $r\ge2$; let $f$ be a polynomial of the form $f(X) = g((X+1)^r)$, which is not a $q$-th power. Let $\omega$ be a $p$-th root of unity.
Prove or disprove that the ...

**0**

votes

**0**answers

35 views

### Is polynomial chaos expansion interesting to surrogate surface?

I'm currently studying polynomial chaos. I want to use it for approximate surfaces but i'm not sure it's possible ? My surface is recursively defined like this : $$ F(x,t) = \underset y \sum ...

**0**

votes

**0**answers

67 views

### Must the radical of polynomial evaluated at integers be small enough at least once?

Basically I am interested if the radical of polynomial evaluated at integers can be small enough at least once.
Let $f \in \mathbb{Z}[x], \deg(f)>1$ be squarefree. For integer $a$ and $f(a) \ne 0$ ...

**5**

votes

**0**answers

161 views

### Operator connected with Hermite polynomials

For $n \geq 1$, define the following operator $M_n$ on the ring of all polynomials with real coefficients.
$$M_n P(x) = nP(x)^2 - x \int_0^x (P'(t))^2 \, \mathrm{d}t$$
Monomials $x^k$ are mapped to $n ...

**1**

vote

**2**answers

122 views

### Results for resolution of equations in polynomial ring

Is there any reference for resolution of equations in a polynomial ring, such as $x^2+y^2=z^2$ in $\mathbb{C}[t]$? Thanks!

**20**

votes

**1**answer

668 views

### $f(x)$ is irreducible but $f(x^n)$ is reducible

Does there exist an irreducible polynomial $f(x)\in \mathbb{Z}[x]$ with degree greater than one such that for each $n>1$, $f(x^n)$ is reducible (over $\mathbb{Z}[x]$)?

**2**

votes

**0**answers

132 views

### Irreducibility of $x^m-g(y)$

Let $g(y)\in \mathbb{C}[y]$, $ m\in \mathbb{Z}_{\ge 2}$. Are there some results on the irreducibility of $x^m-g(y)$ in $\mathbb{C}[x,y]$?

**4**

votes

**0**answers

125 views

### Bound for the height of equations defining the singular locus of a variety

Fix positive integers $m, n, d$.
In what follows, the height of an algebraic number will mean the absolute multiplicative height.
Let $V \subset \bar{\mathbb{Q}}^n$ be an affine algebraic variety ...

**1**

vote

**0**answers

172 views

### Continuous dependence of the roots of a polynomial on its coefficients

In their article "The roots of a polynomial vary continuously as a function of the coefficients" Gary Harris and Clyde Martin give a topological proof of the well-known theorem that the roots of a ...

**7**

votes

**1**answer

380 views

### Integer valued polynomial through some points with rational coordinates

I asked this question on MSE about 5 months ago, but, even after offering a bounty, I didn't receive any answer, I hope this question isn't too easy for MO.
If we have a set of points $(x_i,y_i)$ ...

**7**

votes

**2**answers

436 views

### How to solve such an optimization problem

I encounter the following optimization problem, but I can't solve it.
Given $N$ variables satisfying $0 \leq x_1 \leq x_2 \leq x_3 \leq ... \leq x_N \leq 1$ and an integer $K$ no large than $N$, find ...

**5**

votes

**1**answer

235 views

### Does $ \text{mult}(R / I) = d_{1} \cdots d_{r} $ imply that $ (f_{1},\ldots,f_{r}) $ is an $ R $-regular sequence?

We define the multiplicity of an $ R $-module $ M $ of dimension $ d > 0 $ to be
$$
\text{mult}(M) \stackrel{\text{df}}{=} \text{LC}(P_{M}) \cdot (d - 1)!,
$$
where $ P_{M} $ denotes the Hilbert ...

**1**

vote

**0**answers

113 views

### 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 ? ...

**3**

votes

**0**answers

314 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 ...

**1**

vote

**0**answers

317 views

### When is a cubic polynomial a cube? [closed]

I've been researching cubes and I'm trying to solve this Diophantine equation over the integers.
$$ax^3 + bx^2 + cx + d = y^3$$where a, b, c, d are parameters for a given $n$. For example, for $n = ...

**1**

vote

**0**answers

54 views

### Decomposition of polynomials with three variables

We use $\bigtriangleup _i$ to denote either multiplication or addition.
Suppose we have a polynomial $P(x,y,z)$ over some algebraic closed field such that:
There are $Q(x), W_1(x,y),W_2(x,z)$ ...

**14**

votes

**6**answers

2k views

### Proofs of the Chevalley-Warning Theorem

A well known proof of the Chevally-Warning Theorem is the one listed on wikipedia: http://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem
Are there any other proofs of this, or ...

**3**

votes

**1**answer

119 views

### Is $\mathbb R[x,y]_+$ countably generated as a quadratic module?

Let $\mathbb R[x,y]_+$ denote the set of positive polynomials in two variables. My problem can be stated as follows:
Does there exist a countable set $M\subseteq \mathbb R[x,y]_+$
such that ...

**8**

votes

**1**answer

412 views

### Gaps between roots of trigonometric polynomials

[Cross-posted from Math.SE because I got no responses there.]
Given a polynomial in $e^{\mathrm{i}k t}$ of the form
$$ p(t) = \sum_{-n\leq k\leq n} c_k e^{\mathrm{i}k t} $$
with $\bar c_{-k} = c_k$, ...

**9**

votes

**2**answers

511 views

### Can there be a power basis for a totally real field of high degree?

A number field $K$ is said to have a power basis if there is an $\alpha \in K$ such that the full ring of integers $O_K$ is the $\mathbb{Z}$-linear span of ...

**14**

votes

**1**answer

834 views

### 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.

**2**

votes

**0**answers

104 views

### Existence of roots of high order polynomial over finite fields

I want to solve the following question:
Consider a polynomial $f(x)=a_0+a_1*x^{e_1}+a_2*x^{e_2}+\cdots+x^{e_m}\in F_p[x]$ where $p$ is a prime such that $\log(p)\sim m$ and $e_m\sim 2^m$, I want to ...

**12**

votes

**3**answers

445 views

### Existence of solutions of a polynomial system

Fix $k \in \mathbb{N}$, $k \geq 1$. Let $p \in [0,1]$ and $x = (x_0, \ldots, x_k)$ be a $(k+1)$-dimensional real vector, and define
$$S(p,x) = -x_0^2 + \sum_{i=0}^k {k \choose i} p^i (1 - p)^{k - i} ...

**4**

votes

**0**answers

229 views

### Polynomial dynamical systems

The question is somewhat related to the theory of permutation polynomials. Let $\mathbb{F}_p$ be a finite field of $p$ elements ($p$ is prime) and $\mathcal{V} = \mathbb{F}_p^2 = \{ (t_1,t_2)\::\: ...

**0**

votes

**1**answer

68 views

### Fitting a quadratic using regression when the y-intercept needs to be 0 [closed]

I'm trying to fit a quadratic $a_0 + a_1x + a_2x^2$ by Polynomial Regression:
$$
\begin{pmatrix}
n & \Sigma x_i & \Sigma x_i\\
\Sigma x_i & \Sigma x_i^2 & \Sigma x_i^3\\
\Sigma ...

**1**

vote

**2**answers

107 views

### Estimate maximal coefficient of a polynomial from a circle containing all roots

Suppose I have a polynomial
$$
p(x)=\sum_{i=0}^n p_ix^i.
$$
For simplicity furthermore assume $p_n=1$.
As it is well known we may use Gershgorin circles to give an upper bound for the absolute ...

**6**

votes

**0**answers

229 views

### Factors of the polynomial $X^n-a$

I am interested in the polynomial $X^n-a$ in $\mathbb{Q}[X]$, for some $a\in \mathbb{Q}^*$, and would like to know the irreducible factors of it. Is there something in the literature which gives a ...

**2**

votes

**0**answers

205 views

### Computing the chromatic polynomial of graph modulo $x-3$

The chromatic polynomial of graph $P(G,x)$ is univariate
polynomial which counts the number of colorings of $G$
with $x$ colors for natural $x$.
Graph is not $k$ colorable iff $P(G,k)=0$.
The ...

**9**

votes

**1**answer

286 views

### Families of quintics in $\mathbb{Q}[x]$ with Galois group $A_5$

Theorem. The Galois group of a quintic polynomial $f\in\mathbb{Q}[x]$ is $A_5$ if and only if its discriminant is a rational square and its Weber sextic resolvent has no rational root.
Question. What ...

**1**

vote

**1**answer

71 views

### seeking reference on a theorem about sufficient conditions for an entire function with real coefficients to have only real zeros

I am seeking reference(s) on the following theorem about sufficient conditions for an entire function with real coefficients to have only real zeros.
Theorem:
Let $f_n(z)=\sum_0^n a_m z^m$ (with ...

**1**

vote

**0**answers

36 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 $k$ ...

**3**

votes

**2**answers

285 views

### 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 ...

**6**

votes

**0**answers

168 views

### Find a symmetric polynomial with a projection divisible by a known polynomial

Consider the polynomial $Q$, a homogeneous quartic in seven variables:
$$ Q(R, s_1, s_2, s_3, s_4, d_1, d_2) = \\
...

**1**

vote

**0**answers

211 views

### R[[X]] flat as a R[X]-module?

I assume $R[X]\rightarrow R[[X]]$ is not flat in general, but I was wondering if any conditions on a commutative ring $R$ are known such that $R[[X]]$ is flat as a $R[X]$-module.
Would $R$ noetherian ...

**4**

votes

**0**answers

41 views

### What is the probability of interpolating the Tutte polynomial of a planar graph from the values at the two hyperbolas?

The Tutte polynomial
is a bivariate polynomial with positive integer coefficient which is a graph
invariant and can be defined recursively.
Evaluating it is $\#P$-complete even when restricted to ...

**0**

votes

**0**answers

48 views

### Closed-for expression for Newton-Girard symmetric polynomials with 0/1 variables

There are $n$ Bernoulli $s_i\in\left\{0,1\right\}$, $i=1,...,n$ with equal marginals $\Pr(s_i=1)=\theta$ $\forall i$ so that E$(s_i)=\theta$. Their standardized mean deviations are
\begin{equation*}
...

**4**

votes

**0**answers

154 views

### The equation $f(x)=f(a^n)^k$ always has a solution in $\mathbb{Q}$ [closed]

Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ ...