**2**

votes

**0**answers

95 views

### Uniqueness of the maximum derivative of a rational function

This may seem like an elementary question, but bear with me; you'll find that it is actually quite hard. Consider the function
$$f(x)=\frac{a_nx^n}{\sum_{i=0}^n a_ix^i}$$
with all $a_i\geq 0$ and ...

**3**

votes

**1**answer

129 views

### Real points of zero-dimensional real algebraic varieties

There have been a number of discussions of zeros of random polynomials here (the most recent being: Why do roots of polynomials tend to have absolute value close to 1?).
Here is a closely related ...

**1**

vote

**1**answer

72 views

### Integrals involving trigonometric functions and polynomes

Let $P(x)$ be a real polynome. Specify all such $P(x)$ that one of the next integrals converge:
$$
\int_0^{\infty} sin(P(x))dx, \int_0^{\infty} cos(P(x))dx ?
$$
Among special cases are such ...

**3**

votes

**1**answer

168 views

### A number array related to colored necklaces and the primes

I stumbled upon entry OEIS-A208535 on the enumeration of certain kinds of colored necklaces and noticed that the integers for the odd prime rows of the table there seem to be given by the Moreau ...

**1**

vote

**1**answer

325 views

### A question on polynomials

Let $D=\{0,1\}$.
Are there infinite values of $n\in \Bbb N$ for which there is a polynomial $p(x_1,x_2,\dots,x_n)\in \Bbb R[x_1,x_2,\dots,x_n]$ and a rational function $r(x_1,x_2,\dots,x_n)\in \Bbb ...

**0**

votes

**1**answer

158 views

### Discriminant of a polynomial in two variables

I want to compute the discriminant of the following polynomial
$$
F(X,Y)=X^mY^n+\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}c_{ij}X^iY^j.
$$
Here the discriminate means the equation $D(c_{i,j})$ in the variables ...

**2**

votes

**1**answer

148 views

### Rational functions and polynomials evaluated on a set of points

Let $S$ be a collection of points on the real line.
Let $\{x_i\}_{i=1}^n$ take values in $S$.
Consider a polynomial $p(x_1,x_2,\dots,x_n)$ over $\mathbb R[x_1,x_2,\dots,x_n]$ of degree $d$ which ...

**6**

votes

**1**answer

121 views

### For a Sum-of-Squares cost functions J(x) is it true that J(x)-j* is also SOS?

For polynomial optimization problems the sum-of-squares theory and Lasserre relaxation hierarchy provides a theoretically handy way of getting the solution. There are also results saying that finite ...

**0**

votes

**1**answer

74 views

### About the maximum degree of multivariate polynomial interpolation

It is well known that in the univariate case, to interpolate $k$ points in $\mathbb{R}$, we need to use a polynomial of degree $k-1$.
My question is about multivariate polynomial interpolation in ...

**1**

vote

**0**answers

116 views

### Rings of algebraic integers as quotients of polynomial rings

The ring of integers $\mathcal{O}_K$ of a number field $K$ is always isomorphic to some ring of the form $\mathbb{Z}[x_1, ..., x_r]/\mathfrak{p}$, where $\mathfrak{p} \subset \mathbb{Z}[x_1, ..., ...

**4**

votes

**1**answer

360 views

### Why are the angular differences of these random complex polynomial coefficients almost constant?

This is based on მამუკა ჯიბლაძე's (not-)answer here. I guess it is better to make up a new thread for it.
Let me repeat the setup here: We consider polynomials whose complex roots are randomly ...

**7**

votes

**0**answers

191 views

### Reciprocal polynomials with roots off the unit circle

A polynomial is called reciprocal if its coefficients read forwards are the same as those read backward - there is an obvious translation of reciprocal polynomials into cosine polynomials (since a ...

**3**

votes

**1**answer

72 views

### Multivariate polynomial interpolations

I have a multi-variate, continuous function from $R^n$ to $R$, which I can query for its output for any input. I would like to create an interpolation of that function by sampling a subset of the ...

**0**

votes

**0**answers

71 views

### Algorithm for finding irreducible polynomials in finite field extensions

Let $K(\alpha_1,\ldots,\alpha_n)/K=\tilde{K}/K$ be a finite field extension and suppose we know $\text{irr}(\alpha_1,K)(x),\ldots,\text{irr}(\alpha_n,K)(x)\in K[x]$. Suppose also that we have a basis ...

**216**

votes

**15**answers

31k 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

45 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

147 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

134 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

256 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

87 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

93 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

588 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

203 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

473 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

368 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

165 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 ...

**1**

vote

**0**answers

129 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.
Show that the polynomial ...

**0**

votes

**0**answers

31 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

66 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

151 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

117 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

607 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

128 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

123 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

152 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

362 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

411 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

227 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

107 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 ? ...

**1**

vote

**0**answers

254 views

### A question on partial fraction decompositions

This question concerns a mapping from the poles of a rational function to its partial fraction decomposition coefficients. We assume that the rational function is the inverse of a polynomial of degree ...

**1**

vote

**0**answers

295 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

53 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 $Q(x), W_1(x,y),W_2(x,z)$ ...

**6**

votes

**4**answers

915 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

105 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

364 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

463 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

790 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

90 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

425 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} ...