**3**

votes

**0**answers

174 views

### When does composing polynomials reduce the degree?

Let $\mathbb{F}$ be the field of size 2. For a function $f : \mathbb{F}^n \to \mathbb{F}$, let $d(f)$ be the smallest integer such that there exists a degree-$d(f)$, $n$-variate, multilinear ...

**3**

votes

**0**answers

224 views

### Bounding the degrees in a Bézout relation for integer polynomials

Let $A$ and $B$ be two polynomials in $\mathbf Z[X]$ which generate $\mathbf Z[X]$, that is assume that there exist polynomials $U$ and $V$ in $\mathbf Z[X]$ such that
$$
A \cdot U + B \cdot V=1.
$$
...

**4**

votes

**4**answers

294 views

### Deceptive linear algebra problem

Does anyone recognize this problem? There are $2p$ equations and $2p$ unknowns, and it feels like a classic, but I've never encountered it before:
Given $d_1, d_2, \ldots d_p$, find $a_1, a_2, \ldots ...

**6**

votes

**0**answers

241 views

### Zeros of polynomials related to Jensen polynomial associated with Riemann xi function $\xi(x)$

We encountered polynomials defined by the recursive relations for the coefficients $b_k>0$ as defined below:
$$p_{n}(x)=\sum_{k=0}^{n}\binom{2n}{2k}b_k x^k$$
...

**6**

votes

**2**answers

203 views

### Generalized Characteristic Polynomial with Unimodular Roots

Let us define a diagonal matrix $\mathbf{D}(z) = diag(z^{m_1}, \dots, z^{m_N})$ with $z\in\mathbb{C}$ and positive integers $m_1, \dots, m_N$.
The generalized characteristic polynomial of a matrix ...

**3**

votes

**1**answer

113 views

### Polynomials with Unique Critical Value

My question is extremely simple to state: I am looking for a characterization of multivariate complex polynomials $f$ such that $f(Sing(f))=\{0\}$. My motivation is that I recently read somewhere that ...

**10**

votes

**2**answers

595 views

### Upper density of the set of $n$'s such that $p(n)$ is prime, where $p$ is polynomial

The starting point for this question is the following (false) statement
$\forall n\in \mathbb{N} (n^2 + n + 41 \text{ is prime}).$
Given a polynomial function $p:\mathbb{N} \to ...

**0**

votes

**0**answers

14 views

### Decompose a multivariate polynomial into a permutation on $F_{2^n}$ and an affine transformation

Let $S(x_1,...,x_n)=(y_1,...,y_n)$ be a secret permutation on $F_{2^n}$. $L$ is a secret $F_{2^n}\rightarrow R^{m}$ affine tranformation. $m$ can be smaller than $n$, while $n$ is ususally less than ...

**12**

votes

**2**answers

414 views

### Number of nonzero terms in polynomial expansion (lower bounds)

Let $f(x) = a_1x^{z_1} + a_2x^{z_2} + \cdots + a_kx^{z_k}$ be a polynomial with coefficients $(a_1, \ldots, a_k) \in \mathbb{F}_q^*$ and $z_i$ are distinct positive integers. If I need to compute the ...

**10**

votes

**1**answer

401 views

### When is the image of an integral polynomial contained in the image of another?

Suppose $f$ and $g$ are polynomials with integral coefficients and $f(\mathbb Z)\subset g(\mathbb Z)$. Is there any relation between $f$ and $g$?
For instance, this happens if $f=g\circ h$ for some ...

**1**

vote

**0**answers

63 views

### Lipschitz-like behaviour of quartic polynomials [migrated]

I have observed the following phenomenon:
Let the biquadratic $q(x)=x^4-Ax^2+B$ have four real roots and perturb it by a linear factor $p(x)=q(x)+mx$, so that $m$ not too large with respect to ...

**2**

votes

**1**answer

397 views

### What is the rate of convergence? [closed]

How quickly does the series defined by $$x_0 = 0, \ x_{n+1} = \frac{x_n^2+1}{2}$$ converge to $1$?

**0**

votes

**0**answers

86 views

### Determing signs of Taylor coefficients in entire functions

This is a continuation of Determining when combinatorial sums are zero
Suppose $f(x)$ is an entire function approximated by polynomials with only negative real zeros. Suppose further that ...

**1**

vote

**2**answers

182 views

### Growth of the size of iterated polynomials

I have been working independently on a project but now I am stuck and need to seek an expert's wisdom for a part of it. I am basically looking for theorems related to growth of the size of ...

**7**

votes

**1**answer

216 views

### Another name for coin-flipping polynomials

In his paper Functions arising by coin flipping (section 4), Johan Wästlund coined the term "coin-flipping polynomial" for polynomials that arise in connection with observing a finite number of coin ...

**5**

votes

**3**answers

128 views

### Do the bounded isophase lines of a complex polynomial $f$ through the zeroes of $f’$ define a spanning tree?

Let $f: \mathbb{C} \to \mathbb{C}$ be a polynomial and let $\arg(f(z))$ be the phase of $f(z) = | f(z)| \exp(\mathrm{i} \arg(f(z)))$. The zeroes of $f'(z)$ are saddle points of $\arg(f(z))$, i.e. ...

**11**

votes

**1**answer

330 views

### Orthogonal polynomial under linear transformation

Let $M_n(x) = x^n$ be the standard monomials. The binomial formula allows one to expand $M_n(ax+b)$ as a linear combination of $M_k(x)$, for $k \leq n$, giving
$$
M_n(ax+b) = (ax+b)^n = \sum_{k=0}^n ...

**12**

votes

**2**answers

454 views

### Infinitely many irreducible polynomials of the form f(X^2) + X mod 3?

Are there infinitely many polynomials $f \in \mathbb{F}_3[X]$ for which $f(X^2) + X$ is irreducible?

**2**

votes

**1**answer

78 views

### Dimension of a certain subspace of univariate polynomials

Let $\mathbb{F}$ be an arbitrary field. For a polynomial $f\in\mathbb{F}[x]$,
we use $Z(f)$ to denote set of roots of $f$ in $\mathbb{F}$. Let $S$
and $T$ be sets of elements of $\mathbb{F}$ of size ...

**9**

votes

**1**answer

197 views

### Is an explicit $c$ known to lead to a noncomputable Julia set?

Braverman & Yampolsky have shown that there exist noncomputable Julia sets,
i.e., there exist $c \in \mathbb{C}$ such that the Julia set of $f(z) = c + z^2$
is not computable.
"A set is ...

**17**

votes

**1**answer

478 views

### Minimum value of $|p(1)|^2+|p(2)|^2 +…+ |p(n+3)|^2$ over all monic polynomials $p$

Let $n$ be a positive integer. Determine the smallest possible value of $|p(1)|^2+|p(2)|^2 +...+ |p(n+3)|^2$ over all monic polynomials $p$ of degree $n$.
This question was proposed (problem ...

**2**

votes

**0**answers

76 views

### Rank of a particular matrix

Denote $P_{2n}$ to be collection of homogeneous total degree $2$ real polynomials in exactly $2n$ variables such that coefficient of every monomial is either $1$ or $-1$.
Split variable set into ...

**0**

votes

**1**answer

173 views

### Find all possible rational pairs of a parametric sextic and all cases where it is reducible for either parameter

Find all possible rational solutions pairs $(z,a)$ of the equation
$a^6 + a^4 (-18368 + 9184 z - 2912 z^2) + a^2 (61702144 - 61702144 z + 36814848 z^2 - 10694656 z^3 + 1748992 z^4) - z^2 ...

**0**

votes

**0**answers

42 views

### Algebraic independence in normed spaces

A set of $n$ points in $\mathbb{R}^2$ is algebraically independent over $\mathbb{Q}$ if there is no polynomial dependency among the $2n$ coordinates.
A result (Lemma 3.3) from "Globally linked pairs ...

**-1**

votes

**2**answers

320 views

### What conditions imply that a function over $\mathbb{Z}$ is a polynomial? [closed]

How would one prove that a function is a polynomial? I can't seem to find anything about this on the internet. I would like to know if there are any unique properties that only polynomials can ...

**-1**

votes

**1**answer

57 views

### Question on real polynomial in projective space [closed]

Hi all I was given this question and desperately in need of help as it is part of my graduate studiess I know it is true but my instructor told me to find the right way to do it and I am really ...

**1**

vote

**0**answers

60 views

### About properties of polynomials with common interlacing

Say $\{a_1,a_2,..,a_n\}$ and $\{b_1,b_2,...,b_n\}$ be the real roots of two monic polynomials of degree $n$ which have a common interlacing. (say I have arranged the roots in increasing order)
Can ...

**0**

votes

**0**answers

76 views

### How can I keep the roots of f(x)^n+g(x)^m far away from the roots of f and g?

More specifically, suppose for example I have $h(x)=\sum_{i=1}^k (x-i)^{d_i}$. Can I get any handle on the roots of $h(x)$?
Can I somehow guarantee that the roots of $h(x)$ are not arbitrarily close ...

**6**

votes

**1**answer

337 views

### An S-lemma for polynomials of degree 4 in three variables

Might the following be true:
Let $p,q\in\mathbb{R}[x,y,z]$ be homogeneous polynomials, with $\deg(p)\leq 4$ and $\deg(q)= 2$. Suppose $q(x,y,z)>0$ for some $x,y,z\in\mathbb{R}$. Then the ...

**1**

vote

**1**answer

73 views

### Convergence of sequence of polynomials defined by boundary conditions

I'm sorry if my question sounds trivial, but analysis is not my field.
Consider the interval $[a,b]\subset \mathbb{R}$. On $[a,b]$, for every $n\in\mathbb{N}$, $n\ge 3$, I define the polynomials ...

**36**

votes

**0**answers

2k views

### How to prove this polynomial always has integer values at all integers?

Let $m$ be any positive integer.
$$
P_m(x)=\sum_{i=0}^{m}\sum_{j=0}^{m}{x+j\choose j}{x-1\choose j}{j\choose i}{m\choose i}{i\choose m-j}\frac{3}{(2i-1)(2j+1)(2m-2i-1)}.
$$
Question: $P_m(x)$ always ...

**2**

votes

**0**answers

69 views

### The significance of the Parvaresh-Vardy curve

Even though this seems like a computer science question, it is purely mathematical and concerns polynomials and curves over finite fields.
Consider the Parvaresh-Vardy list decoder.
As I understand ...

**-6**

votes

**1**answer

230 views

### Quintic Equation [closed]

Can we solve the following polynomial quintic equation by radicals
x^5 + x^4 = 1
I found one real root which is algebraic solution (no approximation method ...

**4**

votes

**1**answer

124 views

### Motzkin polynomials and enumeration of chord diagrams

On page 12 of the paper Enumeration of chord diagrams on many intervals and their non-orientable analogs" by Alexeev, Andersen, Penner, and Zograf is a list of polynomials which are a refinement of ...

**10**

votes

**3**answers

569 views

### About the prime divisors of values of polynomials

Let $P(x)$ be a polynomial having integer coefficients (and degree $\geq 3$), and let $p_1<p_2<\dots$ be the prime divisors occurring in the set of values $\{P(n):\ n\in\mathbb{Z}\}$.
Is it ...

**4**

votes

**3**answers

91 views

### counting complex roots which are root of unity times a real number

Let $p(x)$ be a monic polynomial over the integers. I want to count the number of roots which have the form $\zeta \cdot r$ where $\zeta$ is a root of unity and $r$ is a real number.
To count the ...

**4**

votes

**1**answer

76 views

### How to realize any non-crossing matching as $\mathrm{Re}[p(z)]=0$

Asymptotically any polynomial is $p(z) = z^n + O(z^{n-1})$. Therefore $\mathrm{Re}[p(z)]= r^n \cos(2\pi i \theta)$ which vanishes at $\theta = \frac{(k+ \frac{1}{2})\pi}{n}$. Those $2n$ line ...

**0**

votes

**0**answers

56 views

### An additive question on polynomials

Consider $S\cup T=\{0,1\}^n$ where $S\cap T=\emptyset$.
Consider real multilinear (only monomials of form $x_ix_jx_k$) polynomials $P,Q$ such that:
$$Q(S)=0\quad Q(T)\neq0\quad P(S)\neq0\quad ...

**2**

votes

**2**answers

292 views

### Can one estimate the distribution of eigenvalues of a matrix by its Cauchy/Stieltje transform?

Given a real symmetric $n$ dimensional matrix $A$, with eigenvalues $\lambda_i$ I am defining its Cauchy transform as the function, $f_A(z) = \sum_i \frac{1}{z-\lambda_i}\,$
Is there any information ...

**4**

votes

**1**answer

112 views

### Best polygonal approximation to a polynomial $\pm$ c

Let a planar region $R$ be defined
by the vertical range bounded by
a polynomial $f(x) \pm c$ with $c>0$ a constant,
and with $x$ varying between the smallest and largest
roots of $f(x)$.
For ...

**2**

votes

**1**answer

137 views

### Chebyshev Polynomials

Given
$$-\frac{1}2<a<\alpha<0<\beta<b<+\frac{1}2$$
$$+\frac{1}2<c<\gamma<1<\delta<d<+\frac{3}2$$
I want to find a polynomial $f(x)\in\Bbb R[x]$ such that ...

**9**

votes

**3**answers

352 views

### Computing minimal polynomials using LLL

I would like to use the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL algorithm) to compute the minimal polynomial of a (real) algebraic number $\alpha$ from a decimal approximation ...

**7**

votes

**2**answers

261 views

### When is $f(x^d)$ irreducible?

Let $f(x)$ be an irreducible polynomial of degree $n$ over a finite field $\mathbb F_p$. What can we say about $f(x^d)$? When is it irreducible ?

**8**

votes

**1**answer

341 views

### The space of polynomials with all real roots

The question stems from an attempt to answer a question of David Speyer. Let $R \subseteq \mathbb{R}^n$ be the space of coefficients of all polynomials of degree $n$ whose all roots are real, i.e.
$R ...

**7**

votes

**0**answers

196 views

### Theorems proved using combinatorial nullstellensatz that have no other known proof

Alon's (or Alon and Tarsi's?) combinatorial nullstellensatz is a powerful algebraic tool with many applications in combinatorics and number theory. See this, this, this and this mathoverflow question. ...

**35**

votes

**4**answers

2k views

### The sum of squared logarithms conjecture

I am searching for the first proof of (or counterexample to) the following conjecture.
(The sum of squared logarithms conjecture)
For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , ...

**21**

votes

**0**answers

432 views

### Finding a path through real rooted polynomials

This is a lemma I wanted in order to solve Patrizio Neff's conjecture. It turned out to be the wrong way to think about it, but I am still curious if it is true.
Let $z^n+a_{n-1} z^{n-1} + \cdots + ...

**0**

votes

**1**answer

132 views

### When can two Cauchy transforms intersect?

Given two polynomials $p$ and $q$ over reals and being guaranteed that both have all roots real I want to know if there is any characterization of the solutions of the equation $\frac{p'}{p} = ...

**1**

vote

**0**answers

51 views

### Multivariable polynomial interpolation via evaluations from entrywise powers of a point

I am interested in multivariate polynomial interpolation. Within computational complexity theory, I use it to create efficient reductions between counting problems. In the univariate case, there is ...

**0**

votes

**0**answers

91 views

### Polynomial generated with primitive element modulo p

This question is equivalent to the question "Normal basis in cyclotomic number fields" that I asked recently. I am posing this question because maybe in this format somebody can have an answer:
Let ...