**1**

vote

**0**answers

93 views

### approximation of rational functions

Suppose $\hat{p}/\hat{q}$ and $p/q$ are two rational functions where $p,q,\hat{p},\hat{q}$ are of degree $n$. Suppose they satisfy that $|p(z)/q(z) - \hat{p}(z)/\hat{q}(z)| < \epsilon$ for any $z$ ...

**0**

votes

**1**answer

105 views

### Functions of several variables over finite fields [closed]

For a finite field $F$ any function $f\colon F\to F$ is given by a polynomial. My question is what happens when we are given a function of two or more variables? Is this necessarily a polynomial ...

**7**

votes

**0**answers

135 views

### Has anyone seen these binary trees (Catalan-type related to the Gegenbauer polynomials and Motzkin paths)?

The OEIS entry A121448 enumerates binary trees with $n$ edges and $k$ vertices with outdegree 1.
Has anyone seen these trees?
The o.g.f. for this entry, $G(x,t)$, is essentially a discriminant ...

**1**

vote

**0**answers

100 views

### Boundary of pseudospectra

Suppose:
$B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$
${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...

**1**

vote

**1**answer

206 views

### Methods for searching for prime generating polynomials

I am currently using a non-systematic, pseudo-random method for finding prime-generating polynomials, based on the Bateman-Horn method for finding likely candidates, and then narrowing down. I have ...

**0**

votes

**1**answer

93 views

### approximation of products of polynomials

I am wondering whether the following can be proved:
Suppose $p(z)$ and $q(z)$ are polynomials of degree $n$ with real coefficients and leading coefficient 1. Moreover, they have quite different ...

**1**

vote

**1**answer

115 views

### When is a homogeneous polynomial an inner function on the unit torus?

What is the necessary and sufficient condition(s) for a homogeneous polynomial on torus(bidisk) to be an inner function? More precisely I want to know when absolute value of a homogeneous polynomial ...

**5**

votes

**2**answers

162 views

### Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian

This is a question related to the statistical model behind independent component analysis (ICA).
We assume that $Z \sim N(0,1)$. Our goal is to construct a random variable $X$ that follows a ...

**0**

votes

**0**answers

35 views

### finite orbits of transformations of the rational function field

Let $K : = k(x_1, x_2, \cdots, x_t)$ be the rational function field and consider the transformation $\tau$ of $K$ defined by $u \mapsto \frac{\alpha(u) + b}{a}$, where $a \ne 0 , b \in k$ and $\alpha$ ...

**0**

votes

**0**answers

50 views

### Is there any way to approximate the largest root of a polynomial? [duplicate]

I know Newton's method will always give a root, but is there a modified version of the method that will always give the largest root?

**1**

vote

**1**answer

127 views

### Is there a limit definition for the roots of a polynomial with arbitrary degree? [closed]

I know there's no general formula for all the roots of a polynomial with a degree greater than 4, but is there some sort of limit (or other) definition to calculate the roots (particularly the largest ...

**2**

votes

**0**answers

125 views

### Polynomials with some roots whose product is 1

I asked this question in this post but have not got a full answer. So I post it again on MO.
Consider the complex coefficient polynomial equation
\begin{eqnarray}
x^n-\left(a_1+\binom{n}{1}\right)x^{...

**3**

votes

**1**answer

203 views

### Polynomial factoring over finite fields

What is known in general about the complexity of factoring polynomials over finite fields?
For instance given $\Bbb F_q$ where $q=p^n$ and total degree $d$ polynomial in $m$ variables what can we say ...

**31**

votes

**3**answers

1k views

### Polynomials with the same values set on the unit circle

Assume that $P(z)$, $Q(z)$ are complex polynomials such that $P(S)=Q(S)$, where $S=\{z\colon |z|=1\}$ (equality is understood in the sense of sets, but I do not know the answer even for multisets). ...

**1**

vote

**1**answer

97 views

### Ternary cyclotomic polynomials with $n=15r$

Let $n=15r$ where $r>5$ is an odd prime number. If $r\!\!\! \mod 15 \equiv w$ then is it true that $\Phi_{n}(x)$ is not flat whenever $2<w<13$ ? In other words, are the flat ones necessarily ...

**2**

votes

**0**answers

101 views

### Log-concave polynomial is a log-concave function?

A polynomial $\sum\limits_{k=0}^n a_kx^k$ is log-concave if $a_0,\ldots,a_n$ constitute a log-concave sequence. I wonder whether the log-concave polynomial is also a log-concave function with respect ...

**0**

votes

**0**answers

21 views

### Complexity of computing the multivariate Tutte polynomial of clique where each edge have distinct label

The multivariate Tutte polynomial $Z_G(q,v)$
is generalization of the Tutte polynomial and each edge is labelled by
variable $v_e$.
$Z_G(q,v)$ is linear in $v_i$.
Let $G$ be a clique where each edge ...

**1**

vote

**0**answers

21 views

### Complexity of computing the Tutte polynomial of multigraph when the Tutte polynomial of the underlying simple graph is known

Let $G$ be multigraph with $l$ loops and $m$ multiple edges and $G'$ be the
underlying simple graph (loops and multiple edges removed).
Assume the Tutte polynomial of $G'$ is given.
Q1 What is ...

**5**

votes

**2**answers

157 views

### Generalized cycle index polynomial for the symmetric group

The answer to a particular calculation in quantum information theory gives me the following expression:
Given $M$ specific elements of the symmetric group $S_n$, define the polynomial
$$Z_n(\pi_1, \...

**4**

votes

**0**answers

164 views

### Polynomials representing locally constant functions

Let $K$ be a finite field with $p$ elements.
(a) Let $f\in K\lbrack x\rbrack$ be such that (i) $\deg(f)<p$ and (ii) $f(2x) = f(x)$ for $\geq (1-\epsilon) p$ values of $x$ in $K$. What can we say ...

**2**

votes

**1**answer

215 views

### On the divisibility of a certain power sum

Does $1^n + 2^n + \cdots + m^n$ divide $(1+2+ \cdots +m)^n$ for any even integers $m, n\geq 2$ ?.
For $n\leq 4$, the solution easily follows from the relevant identities. For $n\geq 6$, i suspect ...

**5**

votes

**0**answers

163 views

### Is a wild automorphism of $k[x_1,\ldots,x_n]$, $n \geq 3$, necessarily of infinite order?

Let $k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$ of characteristic zero.
When $n=2$, it is known that every automorphism of $k[x_1,x_2]$ is tame, namely, a finite product of elementary ...

**0**

votes

**1**answer

120 views

### Standard rational functions from matrices

In linear algebra we get introduced to standard polynomials that are associated to matrices such as characteristic polynomials and determinants.
What are some of the standard rational functions that ...

**0**

votes

**0**answers

46 views

### Approximation by polynomials with coefficients sum

In this paper Erdos p.1176 remarked that if the coefficients of $f(z):\sum_{v=0}^{n}a_{v}z^v$ are all real,then $\sum_{v=0}^{n}|a_{v}|$ is maximal for $f(z)=\pm T_{n}(z)$,where $T_{n}(z)$ is Chebyshev ...

**2**

votes

**1**answer

76 views

### Distribution of values of quadratic polynomials over a finite field

Let $S$ denote the set of squares in the finite field ${\mathbb F}_p$. Now
let $f(x) \in {\mathbb F}_p[x]$ denote a degree $2$ polynomial, which is not a
square of a linear polynomial. If we consider ...

**0**

votes

**2**answers

501 views

### Solving a system of equations using Gröbner basis

In Sage (or any other package) when using Gröbner basis to solve a system of equations (some of which are non-linear equations) does computing the Gröbner basis for the ideal ID generated by the ...

**2**

votes

**0**answers

131 views

### system of complex number equations

Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that
$$a_1^3+a_2^3+a_3^3+a_4^3=0$$
$$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$
$$(1+2|a_1|^2+2|a_1|^4)a_1+(1+2|a_2|^...

**5**

votes

**1**answer

220 views

### Estimating size of greatest prime divisor of a sequence of integers

Consider the numbers of the form: $$A_n = \prod_{\pm}\left(\pm 1\pm \sqrt{2} \pm \cdots \pm \sqrt{n}\right)$$
where, the product in taken oven all $2^n$ terms with variations in sign. We know such ...

**2**

votes

**0**answers

77 views

### Octahedron and System of trigonometric equations

Could somebody help me to prove the following?
$$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$
$$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$
$$\sum_{k=1}^6 \cos (\phi_k)=0$$
$$\sum_{k=...

**-3**

votes

**1**answer

209 views

### Does differentiation widen, or narrow, the class of functions?

Let $\cal F^k$ be a set of functions, each of class $C^k$,
i.e., both, for every function in $\cal F^k$:
$k^{\textrm{th}}$ derivatives exist, and
are continuous.
Let $D(\cal F^k)$ be the set of ...

**3**

votes

**1**answer

155 views

### Hermite-Kakeya Theorem for entire functions

In a question asked by Bobby Ocean, the following theorem is cited:
Hermite-Kakeya Theorem(for polynomials) - Given two real-valued polynomials, $f$ and $g$, then $f(x)+g(x) r$ has only real zeros ...

**1**

vote

**1**answer

182 views

### Reduced resultant of monic polynomials

Let $f(x)$ and $g(x)$ be coprime monic polynomials in $\mathbf{Z}[X]$ of positive degrees $m$ and $n$ respectively. It seems that in this case their reduced resultant can be obtained from the ...

**8**

votes

**1**answer

575 views

### Sort-of Converse of Kolmogorov Zero-One Theorem

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. The Kolmogorov Zero-One Theorem states that
Suppose we have independent random variables $X_1, X_2, ...$. Then $\forall \ A \in \...

**13**

votes

**2**answers

464 views

### regular polygon question

Let $a_1,a_2,\ldots,a_n$ be distinct points on the complex plane $\mathbb{C}$ and $L$ be a circle in $\mathbb{C}$ such that
$$f(z):=\sum_{i=1}^n|z-a_i|^{2n-2}$$
is constant on $L.$ Could somebody ...

**2**

votes

**2**answers

107 views

### Quick tests to differentiate eigenvalues

Given two real square symmetric matrices $A$ and $B$ are there any quick tests to make sure at least one of their eigenvalues differ without computing the eigenvalues and likely more robust or looking ...

**3**

votes

**0**answers

50 views

### numerical stability of root identification via Newton-Raphson iteration of Stieltjes residue sums

I have asked several questions on math.SE in order to compute numerically the poles of high-degree Padé approximations for $e^{-x}$, because a computation directly from the polynomial ...

**2**

votes

**1**answer

248 views

### On a (possible?) equivalence of Bunyakovsky conjecture

Dirichlet's Theorem on primes in arithmetical progressions is equivalent to the following statement: for all integer numbers $a,b$, where $\gcd(a,b)=1$, there exists at least one prime of the form $an+...

**4**

votes

**0**answers

188 views

### Zeros of polynomials modulo a non-prime

Suppose I have a set $S$ and I want to find a polynomial $p$ such that $p(s) = 0 \mod n$ if $s \in S$, and that it is non-zero modulo $n$ otherwise.
In the literature such an $S$ is sometimes called ...

**4**

votes

**0**answers

85 views

### symmetric systems of polynomial equations

Suppose I have a polynomial $p(x_1,...,x_N)$ in $N$ complex variables, and I wish to solve $p(x_{\pi(1)},...,x_{\pi(N)})=0$ for all permutations $\pi \in S_N$. Clearly this is overdetermined for ...

**2**

votes

**1**answer

93 views

### Pointwise convergence of polynomials to a function on a compact set K that is 1 on some disc D and zero outside D

Motivation of my question: Let $A$ be a bounded selfadjoint operator with spectral measure $E$ and $x$ a vector. Then it is easily seen that the closed linear span of all $A^nx$ ($n\in\mathbb N$) ...

**3**

votes

**0**answers

280 views

### A relation between a ring with its polynomial ring

Let $\{f_i(x)\}_{i\in I}$ be a subset of $R[x]$ where $R[x]$ is the polynomial ring of $R$(a commutative ring with identity). If the ring $R/\langle f_i^2(n)-f_i(n)\rangle_{i\in I, n\in A}$, for every ...

**0**

votes

**2**answers

113 views

### Number of polynomial terms for certain degree and certain number of variables

I would like to calculate the maximum number of polynomial terms given a certain number of variables and a certain degree. eg. given that the number of variables is 2 and the degree is 3, the maximum ...

**5**

votes

**2**answers

105 views

### (Re)construction of a polygon from all inter-vertex distances

For a $n$-polygon in $\mathbb{R}^3$ the set of distances between all pairs of vertices is given. (How) is it possible to reconstruct the geometric structure of the polygone?
Symbolically:
For a set ...

**0**

votes

**2**answers

497 views

### What is wrong with this counterexample to the Weak Bunyakovsky's conjecture and reformulation of Bunyakovsky's conjecture?

From HYPOTHESIS H AND AN IMPOSSIBILITY THEOREM OF RAM MURTY.
On p. 13
BUNYAKOVSKY’S CONJECTURE ( WEAK FORM ). Let $f$ be a polynomial with
integer coefficients and positive leading coefficients ...

**5**

votes

**0**answers

162 views

### Plane real curves such that their intersections with lines are hyperbolic

Let $R$ be an (irreducible) plane real algebraic curve (without isolated points).
Consider Zarissky closure of $R$ in ${\mathbb C}P^1$ (as real variety).
Suppose that $\lambda\in R \Rightarrow\...

**10**

votes

**0**answers

193 views

### Precise relationship between “finite” Fourier analysis and Galois theory in solving the cubic?

Suppose we want to solve$$x^3 - ax^2 + bx - c = 0.$$We know a priori that this can be factored as $(x - r_0)(x - r_1)(x - r_2)$; by Vieta's formulas, we know$$a = r_0 + r_1 + r_2,\quad b = r_0r_1 + ...

**20**

votes

**1**answer

884 views

### Why can the general quintic be transformed to $v^5-5\beta v^3+10\beta^2v-\beta^2 = 0$?

The quintic can be transformed to the one-parameter Brioschi quintic,
$$u^5-10\alpha u^3+45\alpha^2u-\alpha^2 = 0\tag1$$
This form is well-known for its connection to the symmetries of the ...

**12**

votes

**1**answer

1k views

### Is Lehmer's polynomial solvable?

The degree 10 polynomial
$$\displaystyle x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$$
given by D.H. Lehmer in 1933 has the property that its largest real root, $\beta = 1.176280 \cdots$ is ...

**2**

votes

**2**answers

387 views

### Polynomials which always assume perfect power values

Let $f(x)$ be a non-constant polynomial with integer coefficients. It is a well-known result that if $f(n)$ is a square for all integers $n$, then $f$ must in fact be the square of a polynomial (see, ...

**2**

votes

**0**answers

123 views

### Integral representation of the complex homogeneous polynomial $z_1\cdots z_n$

Consider the transform (see e.g., (5.1) in this paper):
\begin{equation*}
\Lambda_\mu(q)(z) := \int_{\Delta_n} q(\zeta)\,\Re\left(\frac{1+\langle\zeta,z\rangle}{1-\langle\zeta,z\rangle}\right)d\mu(\...