**5**

votes

**2**answers

182 views

### $L^{\infty}$ polynomial approximation

In short: For a given smooth or continuous function, how can we obtain the best $L^{\infty }$ approximating polynomial?
Jackson (1911) proved that there is a best approximating polynomial in the $L^{\...

**2**

votes

**0**answers

64 views

### Carlitz factorials and Euler-like series

Let $q$ be a power of a prime $p$. For every $i\in\mathbb N$, one denotes $D_i=\prod_{\substack{h\in\mathbb F_q[T]\text{ monic}\\\deg h=i}}\limits h$. For $n\in\mathbb N$, write $$n=n_0+n_1q+\cdots+...

**3**

votes

**1**answer

207 views

### Polars of algebraic curves and surfaces

I asked this on Math.StackExchange, but received no response, so trying here ...
A paper I'm reading says the following ...
With homogeneous coordinates $\mathbf{x} = [x,y,z,w]$, let $F(\mathbf{x}...

**3**

votes

**0**answers

100 views

### An operator derived from the divided difference operator $\partial_{w_0}$

Some main definitions and basic facts of divided differences:
In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by ...

**3**

votes

**1**answer

71 views

### Prime ideal ramified in extension if and only if certain polynomial divides another one?

Let $k$ be a field of characteristic $ \neq 2$, and let $f \in k[T]$ be a polynomial of degree $\ge 1$ which is square free. Let $K$ be the quadratic extension $k(T)(\sqrt{f})$ of $k(T)$. I know that ...

**3**

votes

**1**answer

131 views

### Non-zero coefficients of primitive polynomials

Let $R$ be the finite field with $q$ elements, and let $m,n\in \mathbb{N}$
be positive integers $\geq 2$. I want to prove that there exists a primitive
polynomial $$F(x) = x^{mn}-\sum\limits_{j=0}^{...

**3**

votes

**0**answers

256 views

### Why is solving polynomial systems NP hard?

Solving polynomial systems is known to be a NP hard problem; however it is not completely clear to me where this complexity comes from.
My interest is in the case of systems of multivariate ...

**1**

vote

**0**answers

56 views

### Uniform limit of m-homogeneous polynomials over compact subsets of a Banach Space

I am trying to solve problem 1.2.A from Mujica's book "Complex Analysis in Banach Spaces".
We denote by $\mathcal{P}_a(^mE;F)$ the space of all $m-$homogeneous polynomials from $E$ into $F$, i.e, the ...

**6**

votes

**4**answers

425 views

### The coefficient of a specific monomial of the following polynomial

Let the real polynomial
$$f_{a,b,c}(x_1,x_2,x_3)=(x_1-x_2)^{2a+1}(x_2-x_3)^{2b+1}(x_3-x_1)^{2c+1},$$
where $a,b,c$ are nonnegative integers.
Let $m_{a,b,c}$ be the coefficient of the monomial $x_1^{...

**3**

votes

**0**answers

75 views

### Sign of bivariate polynomial evaluated over two algebraic numbers

I would like to compute the sign of a bivariate polynomial $f$ evaluated over two algebraic numbers $a$, $b$. The numbers are in "isolating interval representation" meaning that each one is defined by ...

**13**

votes

**0**answers

396 views

### Solving polynomial systems with homotopy. Where is the bottleneck?

I have a polynomial system with $n+k$ unknowns ($n+k$ can be greater than 8), that is known to have a limited number of isolated solutions.
I want to solve this system numerically, but if I plug it ...

**9**

votes

**0**answers

184 views

### Weyl Groups as Galois groups

I am looking for explicit examples (for all positive integers $n \ge 5$) of degree $2n$ even polynomials $f(x)=h(x^2)$ over the field $\mathbb{Q}$ of rational numbers such that the Galois groups of $...

**0**

votes

**0**answers

63 views

### Extremum of the cyclic sum of polynomial ratios (same degree)

I've noticed a few times (probably nothing new) that cyclic sums (assuming $x, y, z > 0$) like:
$\frac{x^2+y^2}{yz} + \frac{y^2+z^2}{zx} + \frac{z^2+x^2}{xy}$,
where in each of the 3 ratios, all ...

**3**

votes

**1**answer

192 views

### system of complex equations

I am working on a system of complex equations The question is the following:
Let $a_1,a_2,\ldots,a_N\in \mathbb{C}$ such that
$$\sum_{j=1}^N \sum_{q=0}^{N-1-k} {N-1 \choose q} {N-1 \choose k+q} |a_j|...

**0**

votes

**0**answers

65 views

### A question about divided differences

I want to ask a question about divided differences.
Let $n\equiv0,1 \pmod 4$ is a positive integer. We know that for any polynomial $f\in \mathbb{Z}[x_1,x_2,\cdots,x_n]$,
$$\partial_{w_0}(f)=\left(\...

**2**

votes

**3**answers

189 views

### Isolating roots of polynomial system

I would like to isolate the regions which contain the roots of a system of two bivariate cubic polynomials.
I thought I would project the solutions onto $x$ and $y$ axis by means of resultant ...

**3**

votes

**0**answers

68 views

### Reducible polynomial compositions

Let $f(x)$ be a polynomial of degree $n\geq 2$ with integer coefficients. Then there always exists a polynomial $g(x)$ such that $f\circ g(x)$ is reducible. Namely, let $g(x)=f(x)+x$; then $f\circ g(x)...

**1**

vote

**0**answers

70 views

### Roots of generalized homogeneous polynomials

A polynomial $P(\xi)$ of $n$ real or complex variables is homogeneous of order $m$ with respect to $\lambda \in \mathbb{Z}_{+}^n$ if $$P(\xi) = \sum\limits_{\substack{(\alpha, \lambda) = m \\ \alpha \...

**1**

vote

**1**answer

335 views

### Independence in mathematics

While trying to think about possible interesting notions of algebraic independance over a skew field, I am wondering where in mathematics appears the notion of being independent, or free over ...

**1**

vote

**0**answers

52 views

### Solving the sextic equation using univariate analytic functions and arithmetic operations

Inspired by the top answer to this MO question, I would like to push the limit of the Hermite-Brioschi-Kronecker theorem. Suppose we only allow solutions to be expressed in terms of basic arithmetic ...

**4**

votes

**0**answers

248 views

### Algebraic curves that enclose and exclude given points in the plane

Q1. Given two finite sets $R,G$ of points in $\mathbb{R}^2$,
$|R|=r$ red points and $|G|=g$ green points,
is it always possible to find a simple closed algebraic curve $C(x,y)=0$
that encloses ...

**1**

vote

**0**answers

30 views

### Particular functional equation for a polynomial

Let $P\in\mathbb C[X]$ be with degree $d\ge1$ and $q>1$.
Can we find polynomials $P_i\in\mathbb C[X]$ and an integer $n\ge1$ such that
$$\sum_{i=0}^nP_i(X^{p_i})P(q^iX)=0,$$
where $p_k$ denotes the ...

**0**

votes

**1**answer

55 views

### Congruences of abelian monoids which can be extended to (ideal) congruences of polynomials

Some weeks ago I asked the same question at [math.stackexchange][1] but I have not gotten any feedback. The flavour of the question (but see the details later) is about whether to understand ...

**5**

votes

**0**answers

166 views

### Solving a Laurent polynomial functional equation

I'm considering a set of functional equations:
For a given $\phi(x)\in\mathbb {Z}[x,\frac {1}{x}] $ with $\phi(x)=\phi(\frac{1}{x}) $,
$f(x)f(\frac {1}{x})+\phi (x)g(x)g(\frac {1}{x})=1, $ where $f(x)...

**1**

vote

**1**answer

67 views

### Positive solutions to simultaneous real quadratic equations

I have a system of $n$ quadratic equations with $n$ unknowns. It can be written as
$diag(x)Ax=1$
$x$ is an $n$-vector, $A$ is $n\times n$, real, symmetric and positive definite, the diagonal ...

**15**

votes

**1**answer

370 views

### Is the number of representations as the sum of two elements of a polynomial sequence always small?

Let $f(x) \in \mathbb{Z}[x]$ be a degree $d>1$ polynomial with integer coefficients. Define
$$r(n) := | \{x,y \in \mathbb{Z} : f(x)+f(y) = n \}|. $$
My question is:
Is it true that $r(...

**4**

votes

**2**answers

298 views

### Inverse of a polynomial map

Ax-Grothendieck Theorem states that if $\mathbf K$ is an algebraically closed field, then any injective polynomial map $P:\mathbf K^n\longrightarrow \mathbf K^n$ is bijective.
Question 1. What does ...

**2**

votes

**1**answer

68 views

### Regularized integral and asymptotic expansion

Let $f:(0, \infty) \longrightarrow (0, \infty)$ be a monotonously increasing function (in fact, a step function) and let $P$ be a polynomial of degree $N$. Suppose I know that for some $k$, the limit
$...

**1**

vote

**0**answers

77 views

### Relation to Ehrhart polynomial with Uniqueness

A set of relative prime, positive integers $A = [a_1, \dots, a_d]$ describe the restricted partition function
$$
p_A(n) = \# \{(m_1,\dots,m_d)\in\mathbb{Z}^d: \textrm{ all }m_j \geq 0, \sum_{j=1}^d ...

**3**

votes

**1**answer

189 views

### Symmetry Group of a Polynomial

Given a polynomial $P \in \mathbb{Z}[X_1,\ldots,X_n]$, is there a poly-time algorithm which computes the group of permutations of variables that leaves $P$ unchanged? (Clearly, the trivial $O(n!)$-...

**2**

votes

**1**answer

73 views

### Division of multivariable polynomials by an ideal

Let $K$ be a commutative field and consider an ideal $I$ of $K[X_1,\dots,X_n]$.
Is there a well behaved "reduction modulo I", in the following sense :
Given a well-ordering $\leq$ on the set of ...

**3**

votes

**1**answer

91 views

### ideal generated from a truncated “real radical-like” set are still real radical?

Suppose $\mathbb{R}[X]$ is the normal multivariate polynomial ring where $X = x_1...x_n$. $\mathbb{R}[X]_t$ is the truncated ring such that $\mathbb{R}[X]_t =\left\{f: f \in \mathbb{R}[X], deg(f) \leq ...

**4**

votes

**2**answers

94 views

### Is there a critical point of a polynomial $f$ within every disc having as diameter the line segment between two zeros of $f$?

Consider a generic complex polynomial $f$ (i.e. one with pairwise distinct zeros [EDITed:] and with pairwise distinct zeros of $f'$) and define for every pair of zeros an open disc which has as ...

**2**

votes

**0**answers

169 views

### Hilbert series of the weight 0 sub-algebra of the algebra of functions on GL(N)

Let $A$ be the algebra of polynomials in $N^2$ variables $x^i_j$, $i,j=1,\dots,N$. It is $\mathbb{Z}^N$ graded, with $\text{weight}(x^i_j)=e_i-e_j$. Here $(e_1,\dots,e_n)$ is the standard basis of $\...

**1**

vote

**0**answers

90 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

101 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

134 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

99 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

204 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

91 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

114 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

149 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

126 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

124 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

200 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

96 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

100 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 ...