**3**

votes

**1**answer

208 views

### Polynomial without roots in a ring

Let $A$ be a commutative ring which is not an integral domain. I try to find a polynomial $P$ of $A[X]$ such that $d°P = 1$ and $P$ admits no root in any ring $B$ such that $A$ is a subring of $B$.

**5**

votes

**1**answer

221 views

### Literature about metapolynomials

We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integer $m$ and $n$, it can be represented in the form
$$f(x_1,\cdots , x_k ...

**3**

votes

**2**answers

153 views

### Unimodular triangulation and Ehrhart polynomials

Let $P$ be a convex lattice polytope. Then it has a polynomial Ehrhart function.
I am interested in what can be said about the Ehrhart polynomial when
$P$ has any of the properties
is integrally ...

**1**

vote

**0**answers

106 views

### polynomial with rational coefficients [closed]

Do you think that the following statement is true? Do you have any idea about the proof?
Let $\; f(x) \in \mathbb{C}[x]$ be a polynomial. If $f(n) \in \mathbb{Z}$ for an infinite number of $n \in ...

**3**

votes

**0**answers

108 views

### Discriminant polynomial generalizing the usual discriminant

I wonder if anybody has seen the following natural polynomial.
Given a monic univariate polynomial $P(z)$ of degree $N$, denote its roots by
$z_1,..., z_N$. Now form a new polynomial $Q(z)$ of ...

**2**

votes

**1**answer

106 views

### Roots of modified polynomials

Consider the following two polynomials:
$$
g=x^3 - x^2 - (c + 2)x + c
$$
and
$$
h=x^3 - x^2 - cx + c
$$
The roots of $h$ are $1$ and $\pm \sqrt{c}$. I am interested in obtaining the roots of $g$, ...

**16**

votes

**2**answers

583 views

### Number of zeros of a polynomial in the unit disk

Suppose $m$ and $n$ are two nonnegative integers. What is the number of zeros of the polynomial $(1+z)^{m+n}-z^n$ in the unit ball $|z|<1$?
Some calculations for small values of $m$ and $n$ ...

**6**

votes

**7**answers

667 views

### Source for roots of matrix polynomials?

A
matrix polynomial
is a polynomial whose variables are square $n \times n$ matrices,
let's say with entries in $\mathbb{C}$, and with coefficients in $\mathbb{C}$.
I am seeking a source of results on ...

**4**

votes

**1**answer

222 views

### Stronger versions of Schwartz-Zippel for random linear subspaces

This is a (self-contained) followup question to http://math.stackexchange.com/questions/380672/analogue-of-the-schwartz-zippel-lemma-for-subspaces.
Let $f : \mathbb{R}^n \to \mathbb{R}$ be a nonzero ...

**2**

votes

**1**answer

134 views

### Lagrange Interpolation and integer polynomials

Suppose that there is a polynomial $P$ with integer coefficients such that $P(x_i)=y_i$ for $i=1,\ldots,n$. Is it true that the result of Lagrange interpolation through the data $(x_i,y_i)$ is a ...

**1**

vote

**0**answers

80 views

### Generalization of the Hermite-Biehler-Kakeya Theorem (2)

This is a follow up to the questions posed in Generalization of the Hermite-Bielher-Kakeya Theorem. Here is an interesting follow up to those comments.
Firstly we remark that: $f(x)+g(x)\cdot w$ is ...

**1**

vote

**0**answers

156 views

### The gcd of coprime polynomials evaluated at integers

Let $p(x),q(x)$ be coprime squarefree polynomials with
integer coefficients.
For integer $n$ is $\gcd(p(n),q(n))$ bounded by
an absolute constant?
In case the answer is negative what is
the fastest ...

**1**

vote

**0**answers

36 views

### Find a common expression (parameterized by y = 1/2, 1 and 2) uniting three (rational) polynomials in k [closed]

I am seeking a common "simple" expression (preferably/presumably a sum of products of linear factors, or sum of products of low-degree factors) uniting the three polynomials (parameterized by ...

**2**

votes

**2**answers

189 views

### Sextic resolvent has no rational root

An irreducible quintic $f(x)\in\mathbb{Q}[x]$, is solvable by radicals if and only if its sextic resolvent $\theta_f (y)=(y^3+py^2+qy+r)^2-2^{10}\Delta(f)y$ has a rational root ($\Delta$ is the ...

**2**

votes

**0**answers

260 views

### Enumerating certain types of permutation polynomials

Given a prime power $q$, I would like to enumerate (preferably up to isomorphism*) all the permutation polynomials $f(x)$ on $K = GF(q^3)$ satisfying the following conditions:
$f(ax) = af(x)$ for ...

**0**

votes

**0**answers

31 views

### number of bipolar orientations to acyclic ones

Given a $k$-regular graph $G$, is there an upper bound on the number of bipolar orientations that $G$ has?
I am trying to show that the number of bipolar orientations is much much lower than the ...

**5**

votes

**0**answers

162 views

### Are the roots of chromatic polynomials plus a fixed constant dense in $\mathbb{C}$?

Alan Sokal proved that chromatic roots are dense in the whole complex plane. I.e., if $P(G;z)$ denotes the chromatic polynomial of a finite simple graph $G$ evaluated at $z \in \mathbb{C}$, then ...

**3**

votes

**2**answers

355 views

### Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$

I’m facing the problem of factoring polynomials of type $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ and the degrees of factors seem to be quite special. For example, according to my experimental results ...

**10**

votes

**5**answers

902 views

### Is $x^p-x+1$ always irreducible in $F_p[x]$?

It seems that for any prime number $p$ and for any non-zero element $a$ in the finite field $\mathbb F_p$, the polynomial $x^p-x+a$ is irreducible over $\mathbb F_p$. (It is of course obvious that ...

**3**

votes

**1**answer

190 views

### Generalization of the Hermite-Bielher-Kakeya Theorem

A stable polynomial is one with zeros in the upper half plane or lower half plane. Interlacing polynomials are polynomials with only real zeros, where between every two zeros of one polynomial lies a ...

**1**

vote

**0**answers

81 views

### Estimating decay of certain trigonometric polynomials

For $p=0,1,2,\dots$ and $n=0,1,2,\dots,$, let $f_{n,p}(z)=\sum_{k=0}^n k^p z^k$ be a sequence of polynomials. Restricted to the unit circle, the functions $g_{n,p}(t):=f_{n,p}(e^{it})$ are ...

**1**

vote

**1**answer

151 views

### Polynomial convex coefficients

Assume we have an arbitrary high order polynomial $$f(L)=1-L\theta_1-L^2\theta_2-L^3\theta_3-...-L^N\theta_N$$ and we know all roots of this polynomial site outside the unit circle. It is obvious that ...

**2**

votes

**1**answer

134 views

### Nilradical and Newton's identities

Let $R$ be a commutative ring with unity such that $n!$ is not a zero-divisor. Let $s_1=\sigma_1,s_2,s_3\cdots$ and $\sigma_1,\sigma_2\cdots$, (convention: if $k>n$, then $\sigma_k=0$) be elements ...

**3**

votes

**2**answers

174 views

### “Degree 3 fields”

I was wondering what was known about fields $k$ having the property that any polynomial
over $k$ of degree $3$ has at least one root in $k$. Does such a field have a special name ?
Is there some kind ...

**17**

votes

**1**answer

1k views

### How to prove that every polynomial in an infinite family is irreducible over Q?

Consider the bivariate polynomial $$p(X,Y) = X^5 - (2 Y + 1) X^3 - (Y^2 + 2) X^2 + Y (Y-1) X + Y^3.$$ For every integer $y \ge 4$, I conjecture that $p(X,y)$ is irreducible in $\mathbb{Q}[X]$. How can ...

**1**

vote

**0**answers

36 views

### Identification of model involving convex polynomials

I want to solve a nonlinear least squares problem on the following form
\begin{equation}
\begin{array}{l}
\min_{\theta,\phi} J(\theta,\phi) &=& \min_{\theta,\phi} \sum_{i=1}^k ...

**4**

votes

**0**answers

159 views

### The Alexander-Conway polynomial: from knots to braids?

The Alexander-Conway polynomial was the first knot invariant to be discovered, as far back as 1923 according to this link. Given that knots can be expressed in terms of quasi-toric braid closures, it ...

**1**

vote

**0**answers

61 views

### Whether r.v. with p.g.f. $\exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)]$ is overdispersion?

When discrete r.v. $X$ is not Poisson distributed and ${\rm{Var}}X,EX < \infty $, I want to know whether r.v. $X$ with p.g.f. $\exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)],({q_i} \in ...

**10**

votes

**3**answers

266 views

### Uniform bound for the number primes $p$ s.t. a polynomial has a root modulo $p$

It is well known that for a given polynomial $f \in \mathbb{Z}[x]$ the number of primes $p$ s.t. $f$ has a root modulo $p$ is infinite. In fact, one can even write down a formula for the density of ...

**9**

votes

**1**answer

684 views

### How small can a totally positive integer be?

Consider a large, fixed $M>2$. For each $n$, let $\alpha_n$ denote the smallest algebraic integer of degree at most $n$, all of whose Galois conjugates lie in the real interval $(0,M)$.
Is there ...

**3**

votes

**1**answer

160 views

### Convergence of the Double Integral of a Polynomial Reciprocal

Let $f \in \mathbb{R}[x,y]$ be a polynomial satisfying the following conditions:
(i) $f(\mathbb{R}^2) \subset [a,\infty)$ where $a>0$;
(ii) $f$ is non-degenerate, in the sense that there isn't a ...

**3**

votes

**1**answer

301 views

### Counting solutions modulo primes

Let $P(x)$ be an irreducible polynomial in $\mathbb{Z}[x]$ of degree $n.$ By $N(k,x)$ we denote the number of primes up to $x,$ such that $P(x)$ has exactly $k$ solutions in $\mathbb{Z}_p.$ Is it ...

**0**

votes

**0**answers

77 views

### Complexity of turning a d-degree polynomial to 2-degree polynomial

For a very simple example,
$(1+x)^4=x^4+4x^3+6x^2+4x+1$ is a 4 degree polynomial, and I want to change it to a 2-degree polynomial by add more variables, for this example, we can simply let $y=x^2$, ...

**1**

vote

**1**answer

207 views

### The Largest Root of Associated Laguerre Polynomial

The Laguerre polynomial $L_n(x)$ is the solution to the Laguerre differential equation
\begin{equation*}
x\,y'' + (1 - x)\,y' + n\,y = 0.
\end{equation*}
The associated Laguerre polynomial ...

**0**

votes

**1**answer

159 views

### k-th largest root in common interlacing polynomials

In their proof of the celebrated Kadison-Singer conjecture, Marcus, Spielman and Srivastava exploited so-called interlacing families which are originally defined for their work on Ramanujan graphs. ...

**1**

vote

**2**answers

187 views

### Under what conditions does the Mittag-Leffler function ${E_{\alpha ,1}}(z),(0 < \alpha < 1)$ has no real zero?

I want to know under what conditions does the Mittag-Leffler function ${E_{\alpha ,1}}(z),(0 < \alpha < 1)$ has no real zero, where
${E_{\alpha ,1}}(z) = \sum\limits_{k = 0}^\infty ...

**8**

votes

**1**answer

193 views

### is there any such result about Bernstein polynomials?

It is well known that for any lipschitz function $f:[0,1]\rightarrow [0,1]$, we can approximate it
by $\sum_{i=1}^n f(i/n) {n\choose i} x^i (1-x)^{n-i}$, and the $L_\infty$ error is $O(1/\sqrt{n})$. ...

**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 ...

**4**

votes

**0**answers

141 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

74 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
...

**3**

votes

**2**answers

285 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

125 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

182 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

574 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

162 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

139 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

260 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 ...

**16**

votes

**2**answers

444 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

627 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

61 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 ...