**17**

votes

**3**answers

2k views

### Where in mathematics do these polynomials appear?

Does anyone recognize the following sequence of polynomials?
$f_0(x) = x-1$
$f_1(x) = x^2-x$
$f_2(x) = x^4-2x^2+x$
$f_3(x) = x^8-3x^4+3x^2-x$
$f_4(x) = x^{16}-4x^8+6x^4-4x^2+x$
$\vdots$
The ...

**3**

votes

**0**answers

125 views

### Correspondence between numerical semigroups and polynomials?

A numerical semigroup $A$ is defined as a subsemigroup of the semigroup $(\mathbb{N},+)$ of the positive integers such that the set $\mathbb{N}\setminus A$ is finite. Equivalently (for a subsemigroup) ...

**3**

votes

**1**answer

68 views

### Counting extrema on a simplex

Let $p(x_{1},x_{2},\ldots,x_{n})=\sum_{i,j=1}^{n}{a_{ij}x_{i}x_{j}}$ be a homogenous multivariate polynomial of degree $2$.
I would like to know how many extrema $p$ has on the standard simplex
...

**2**

votes

**2**answers

213 views

### Factorisation of a biquadratic polynomial

Let $u,v\in\mathbb{Z},$ and let $f=X^4+uX^2+v.$ Let $p$ be a prime number, and let $r\geq 1.$
In a paper I'm reading, one can find the following result.
Proposition. If $f$ is reducible modulo ...

**3**

votes

**0**answers

119 views

### Question related to the number of rational curves on a hypersurface

The following is from a PhD thesis "Algebraic Circle Method" by Thibaut Pugin, which I am currently reading.
Let $k$ be a finite field of order $q$.
Let $\mathfrak{X} \subseteq \mathbb{P}^n_k$ be ...

**7**

votes

**1**answer

160 views

### Inequality for Laguerre polynomials

Let $L_n$ be the $n$-th Laguerre polynomial defined by
$\quad
L_n
(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}(x^n e^{-x}).\quad
$
I want to prove that
$$
\forall n\in \mathbb N,\forall x\ge 0,\quad \sum_{0\le ...

**2**

votes

**1**answer

530 views

### Absolute convergence of logarithm of polynomial with positive coefficient ($\ln G(z) = \sum\limits_{i = 0}^\infty {{q_i}{z^i}} $)

Special problem: Let $G(z)$ be a probability generating function(pgf, the $z$ can be seen as real number or complex number), that is
$$G(z) = \sum\limits_{i = 0}^\infty {{p_i}{z^i}} ,(\left| z ...

**1**

vote

**1**answer

155 views

### Zeroes of a complex polynomial on a sphere as a manifold

Let $ f \in \mathbb{C}[z_1, \ldots, z_n]$ be a polynomial such that $f'(z) \neq 0$ if $z \neq 0$ ($f'$ means $\left( \frac{\partial f}{\partial z_1}, \ldots, \frac{\partial f}{\partial z_n}\right)$ ). ...

**3**

votes

**0**answers

128 views

### Number of rational curves on varieties over finite fields

Let $k$ be a finite field. Let $P_r$ be set of degree $r$ binary forms
over $k$.
We define
$$
\mathcal{M}_r = \{ (x_0, ..., x_n) \in P_r^{n+1} : x_0^d + ... + x_n^d = 0 \text{ and the }x_i \text{'s ...

**2**

votes

**3**answers

181 views

### Multivariate Hermite Polynomials

Let $h_0, h_1, \dots$ be the classical univariate Hermite polynomials, renormalized to have constant norm. Is
$$x\mapsto\prod_{j=1}^n h_{l_j}(x_j), \quad l_j\in \mathbb N$$
a complete orthogonal ...

**15**

votes

**2**answers

425 views

### Minimum number of variables on which a multivariate polynomial depends?

Let $p:F_2^n\rightarrow F_2$ be a multivariate polynomial, let's say of degree 3. (Both the degree and the order of the field could probably be replaced by other constants without affecting this ...

**6**

votes

**2**answers

588 views

### Would such polynomial identity exist? (related to sum of four squares)

Let $f_1,f_2,f_3,f_4,f_5 \in \mathbb{Q}[x]$ be linear and
coprime and not all constant.
Is it possible $ f_1^2+f_2^2+f_3^2+f_4^2=f_5^2$?
I suppose the answer is negative.
If this is possible, ...

**6**

votes

**0**answers

59 views

### Bounding volume of cell in complement of zero set

I am given an integer polynomial $f \in \mathbb{Z}[X_1, \ldots, X_n]$ of bounded degree and bounded coefficient size. The polynomial's zero set partitions $\mathbb{R}^n$ into cells. What I am looking ...

**2**

votes

**0**answers

125 views

### Must the coordinates of a polynomial iteration have about the same size?

Original post. The following statements seem plausible (not to say intuitively obvious), but I do not see how to prove them.
Let us say that a polynomial mapping of $\mathbb{C}^2$ is reducible if ...

**7**

votes

**1**answer

119 views

### Positivity of coefficients of a power series

How does one check for the positivity of coefficients of a rational function,say, for example $\frac{p_1(x,t)}{(1-xt)(1-x^2t)(1-x^3t)}$ where $p_1(x,t) = 1 + tx + 2t^2x^2 - 3x^3t^2 -x^5t^3 - ...

**1**

vote

**0**answers

80 views

### Irreducibility of reflexive compositions of polynomials (II)

Given an unknown polynomial $p \in \mathbb{Z}[x]$, write on the blackboard a finite expression that defines a polynomial $q \in \mathbb{Z}[x]$ by using only constants $1, x, p$ of the ring ...

**2**

votes

**0**answers

156 views

### square-free parts of values of polynomials

Given a polynomial $f(x) \in \mathbb{Z}[x]$ of degree $d$, consider the following three sets:
$$N_1(x) = \#\{k \leq x: f(k) \text{ is square-free}\}$$
$$N_2(x) = \#\{n \leq x: n = f(k) \text{ is ...

**10**

votes

**1**answer

274 views

### Minimize norm of a polynomial around a circle (count the solutions)

I already posted this question at MSE here, but as it received no significative feedback for a while I cross-post it here.
I also noticed a related question here on MO (which does not answer my ...

**4**

votes

**2**answers

220 views

### Finding roots of a recursively given polynomial sequence / eigenvalues of an almost Toeplitz matix

I would like to find the roots of the polynomial sequence given by a recurrence relation as follows:
$V_0(x) = 1-a^2$
$V_1(x) = 1-a^2 - x$
$V_{k \geq 2}(x) = (1+a^2 - x)V_{k-1}(x) - a^2V_{k-2}(x)$
...

**9**

votes

**3**answers

1k views

### An algebraic number is not a root of unity?

This problem is related to my study of the Burau representation of the braid group $B_3$: I was trying to show that certain "congruence subgroups" are of infinite index.
There is an approach that ...

**34**

votes

**2**answers

825 views

### A curious identity related to finite fields

To three elements $a_1$, $a_2$, $a_3$ in the finite field $\mathbb F_q$
of $q$ elements we associate the number $N(a_1,a_2,a_3)$
of elements $a_0\in \mathbb F_q$ such that the polynomial
...

**1**

vote

**1**answer

82 views

### Polynomials are dense in $A_{B(0,1)}$

Let $D(0,1)$ be the disk of center 0 and radius 1 and call $A_{D(0,1)}= \{ f:\overline{D(0,1)} \rightarrow \mathbb{C} : f \text{ is continuous and } f|_{D(0,1)} \text{ is holomorphic} \}$.
Can ...

**0**

votes

**1**answer

267 views

### Generators of cyclic group of finite fields

Let $F$ be a finite field and $f(x)\in F[X]$ be an irreducible polynomial of degree $n$. Let $\alpha$ be a root of $f(x)$. So $E:=F[\alpha]$ is a finite field of order $|F|^{n}$.
We know that ...

**7**

votes

**4**answers

213 views

### Is there an oscillating analog of the Gaussian distribution?

It frequently happens that, in some famillies of polynomials with positive coefficients, the coefficients of large polynomials look like a bell curve and tend to the distribution function of the ...

**12**

votes

**1**answer

360 views

### Probability that random cubic polynomials meet in a square

Let $p_1(x)$ and $p_2(x)$ be cubic polynomials with
random coefficients in $[-1,1]$.
I wanted to compute the probability that $p_1$ and $p_2$
share at least one point within
the square $[-1,1]^2$.
Of ...

**1**

vote

**0**answers

46 views

### Given a multivariate polynomial with even degree, can we find its tightest convex polynomial 'envelop'?

To be specific, given a multivariate polynomial function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ with even degree $2d$, can we construct a convex polynomial function $g$, such that:
$\forall ...

**9**

votes

**1**answer

540 views

### Real-rootedness, interlacing, root-bounds of a sequence of polynomials

Problem: the number $a(n,k)$ is defined by the following recurrence
\begin{equation}
a(n,k)=(k+1)(k+2)\, a(n-1, k)+\frac{(k+1)(k+2)(k+3)}{k} \,a(n-1, k-1),
\end{equation}
with $a(1,1)=1$ and ...

**0**

votes

**0**answers

31 views

### Threshold approximation with positive coefficients polynomials

Consider the family of polynomials $\{p_n\}$, each of degree $n$, such that for every polynomial $p = \sum_{i}c_{i}x^{i}$, $c_i \ge 0$ for every $i$ and $\sum_{i}c_i = 1$.
Let $T$ be the ...

**5**

votes

**2**answers

269 views

### Can we decompose a polynomial into difference of convex polynomials?

Given a multivariate polynomial $p(x_1, ..., x_n)$ on $\mathbb{R}^n$, can we always decompose it into the difference of two convex polynomials? i.e., is there a pair of convex polynomials $f$ and $g$, ...

**2**

votes

**1**answer

116 views

### a second order difference equation related to a real polynomials which seems to have only real roots

I am seeking solutions to the following difference equation:
$$2c_k-c_{k-1}-c_{k+1}=\ln(k+A)-\ln(k+B)$$
where $A>B>0$.
This equation is related to a real polynomial (see here) which I want to ...

**0**

votes

**0**answers

68 views

### unique positive real root fast computation

What is the fastest way to compute the value of the unique positive real root corresponding to the following polynom:
:p(x) = a*x^5 + b*x^4 + c*x^3 + d*x^2 + e*x - f = 0
where a, b, c, d, e, f are ...

**4**

votes

**1**answer

187 views

### A palindromic polynomial and its derivative have the same number of zeros outside the unit circle. Reference?

I am trying to find the original reference for a lemma attributed to Cohn (as in Schur-Cohn method):
Let $A(z)$ be a palindromic or skew-palindromic polynomial, and denote its derivative by ...

**0**

votes

**1**answer

146 views

### Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial

The following question is motivated by the study of a stability border for a robust linear time-invariant control system.
Let us we have an affine family of $n\times n$ matrices with indeterminate ...

**6**

votes

**0**answers

197 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:
$$p_{n}(x)=\sum_{k=0}^{n}\binom{2n}{2k}b_k x^k$$
...

**1**

vote

**1**answer

120 views

### Truncated sums of symmetric polynomials; reference request for an algebraic derivation

EDIT: This is a case of being too wrapped up in a formulation
($e_j,p_i,$ and the like) to try something simple. It did not
occur to me to pull exp to the outside in the weeks I have
stared at this. ...

**2**

votes

**1**answer

204 views

### Powers of linear functions span the space of polynomial functions?

Let $P_n$ be the space of polynomials in $n$ variables over a field of characteristic 0.
I'm very sure that the space spanned by powers of linear functions is the whole space $P_n$.
Anyone can come ...

**0**

votes

**0**answers

38 views

### What is maximum number of m-complex solutions to a order n polynomial (say with real coefficients)?

I know the answer is n^2 for bicomplex numbers. Does anyone know if a general answer has been found for m-complex numbers (
http://en.wikipedia.org/wiki/Multicomplex_number)?

**0**

votes

**1**answer

114 views

### Finding an interpolating polynomial on a set of points such that the polynomial only has extrema on those points

Suppose that we have an arbitrary set of points $S\subseteq\mathbb{C^2}$ and want to find a polynomial $P(X)$ such that $(x,y)\in S\Rightarrow y=P(x)$ and $P'(x)=0\Rightarrow (x, P(x))\in S$. In other ...

**4**

votes

**2**answers

306 views

### Can a polynomial be almost always divisible by a member of a finite set of primes?

Special case of Bunyakovsky conjecture
Let $f(x)$ be non-constant irreducible polynomial with integer
coefficients, no fixed prime factor and positive
leading coefficient. Let $S$
be a finite set of ...

**8**

votes

**1**answer

262 views

### Irreducibility of a class of polynomials

This question is directly inspired by this question. Consider polynomials of the form
$$p(x) = \prod_{i=1}^n(x-i)^2 - d.$$ For which values of $n$ and $d$ is $p(x)$ irreducible? There is a theorem of ...

**2**

votes

**0**answers

218 views

### For which constant $d$ this polynomial is reducible over $\mathbb Q$? [closed]

Let $$f(x)=(x-1)^2(x-2)^2(x-3)^2\cdots(x-2013)^2+2014\tag{1}$$
Prove or disprove: $f(x)$ is reducible over $\mathbb Q$.
See :http://www-irma.u-strasbg.fr/~bugeaud/travaux/PolyaType.pdf
...

**3**

votes

**3**answers

362 views

### Can you efficiently solve a system of quadratic multivariate polynomials?

Given a system of 2nd-degree polynomials, $P=\{p_1,\dots,p_m\}$ where $p_i: \mathbb{R}^n \rightarrow \mathbb{R}$, can you efficiently find $x_1,\dots,x_n\in\mathbb{R}$?
I know that in the general ...

**3**

votes

**0**answers

216 views

### The Poisson-kernel in the plane and polynomials

Let
\begin{align*}
p(z) & = \ \prod_{j = 1}^{n} (z - z_j) \, ; \ |z_j| \ = \ 1 \\
& = \ \prod_{j = 1}^{l+1} (z - z_j)^{M_j}
\end{align*}
be a non-constant complex polynomial with ...

**10**

votes

**3**answers

519 views

### Relating the roots of polynomials to the solution sets of certain functional equations

Consider a functional equation of the following form:
$$\sum_{k=0}^n a_k\,\underbrace{f(f(\cdots f}_{k}(x)\cdots )=0\quad \big(f:\,\mathbb{R}\to\mathbb{R},\;a_i\in ...

**4**

votes

**2**answers

400 views

### Characterizing $\mathbb{Q}[X]$ via a property of its tensor powers

Let $\varphi: \mathbb{Q}[X] \longrightarrow R$ an inclusion of commutative rings. Suppose that the map
$$- \circ \varphi: \operatorname{Hom}_{\mathbb{Q}\operatorname{-alg}}(R, R^{\otimes_{\mathbb{Q}} ...

**10**

votes

**1**answer

251 views

### Reducibility of polynomials maps

Motivated by this question.
Let $f \in \mathbb{Q}[x]$ or$f \in \mathbb{Z}[x]$ .
Consider the sequence $f(x),f(f(x)), \ldots f^n(x)$.
If some $f^k(x)$ is reducible, the rest iterates will be ...

**7**

votes

**2**answers

90 views

### Bounds on coefficients of factors of a multivariate polynomial

Given a multivariate polynomial $F(x, y, ..)$ what is the smallest bound B that can be quickly found such that $|G|_{\infty} \le B$ for all factors $G$ of $F$. (I'm using $|G|_{\infty}$ to denote the ...

**1**

vote

**0**answers

53 views

### Irreducibility of reflexive compositions of polynomials

Let $m$ be a positive integer. For $k = 1, 2, 3, ... m$, fix $g_k(x_1, ..., x_{k + 1}) \in \mathbb{Z}[x_1, ..., x_{k + 1}]$.
For any polynomial $p(x) \in \mathbb{Z}[x]$, let
$P_0(x) = p(x), P_1(x) ...

**3**

votes

**2**answers

341 views

### The holomorphic version of Galois theory

We identify the space of polynomials of degree n with $\mathbb{C}^{n+1}-\mathbb{C}^{n}$, that is an $n+1$ tuple $(a_{n},a_{n-1},\ldots,a_{0})$ with $a_{n} \neq 0$ is identified with $p(z)=a_{n}z^{n} ...

**3**

votes

**2**answers

266 views

### Irreducible polynomials in $\mathbb{Q}_p((X))[Y]$

I'm looking for some criteria for the irreducibility of polynomials with coefficients in $\mathbb{Q}_p((X))$.
In particular, is the polynomial $Y^2+1$ irreducible over $\mathbb{Q}_3((X))$? And how ...