Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ...

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3
votes
1answer
108 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 ...
0
votes
0answers
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 ...
1
vote
0answers
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 ...
6
votes
2answers
190 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 ...
10
votes
1answer
397 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 ...
2
votes
1answer
391 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 $1$?
12
votes
2answers
404 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 ...
0
votes
0answers
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
2answers
178 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 ...
10
votes
2answers
580 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 ...
5
votes
3answers
125 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. ...
12
votes
2answers
451 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?
7
votes
1answer
215 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 ...
9
votes
1answer
196 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
1answer
471 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
0answers
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
0answers
41 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
1answer
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
votes
2answers
319 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 ...
0
votes
0answers
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 ...
1
vote
1answer
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 ...
2
votes
0answers
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 ...
4
votes
1answer
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 ...
4
votes
3answers
90 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
1answer
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 ...
36
votes
0answers
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 ...
0
votes
0answers
55 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 ...
4
votes
1answer
111 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 ...
10
votes
3answers
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 ...
7
votes
2answers
259 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
1answer
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
0answers
195 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. ...
2
votes
1answer
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 ...
-6
votes
1answer
228 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 ...
2
votes
2answers
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 ...
21
votes
0answers
430 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 + ...
1
vote
0answers
58 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 ...
1
vote
0answers
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 ...
35
votes
4answers
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 , ...
0
votes
1answer
131 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} = ...
0
votes
1answer
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 ...
5
votes
2answers
185 views

Continuous map from connected subset of R^n to one of the real zero of an odd degree polynomial whose coefficients are polynoms of the variables

Let take a real multivariate polynomial $P(x_1, \ldots, x_n, y)$ such as the degree of P relatively to the variable $y$ is odd. Thus, for each $X = x_1,\ldots,x_n \in\mathbb{R}^n$, the univariate ...
1
vote
1answer
42 views

Splines linearly independent

Let $N_1:=\chi_{[0,1]}$ be defined as this characteristic function and $N_n:=N_{n-1}*N_1$ then this leads to polynomials with support $[0,n]$. These splines are well-studied click for wikipedia My ...
4
votes
1answer
177 views

Macaulay's example of prime ideals in $\mathbb C[X_1,X_2,X_3]$ having large number of generators

There is a famous example of Macaulay which shows that there are prime ideals of height two in $\mathbb C[X_1,X_2,X_3]$ having at least $l$ generators for any $l\ge 3$. In Macaulay's words, the ...
2
votes
1answer
76 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 ...
1
vote
1answer
216 views

Composition of rational functions

Given a rational function $R\in\Bbb R(x_1,\dots,x_n)$ with multilinear numerator and denominator, is there always a rational function $G\in\Bbb R(x)$ such that $G\circ R\in\Bbb R[x_1,\dots,x_n]$, ...
2
votes
0answers
108 views

What are the general zonal spherical functions for ${\rm SO}(n)/{\rm SO}(n-1)$?

The zonal spherical functions [1] on the sphere $(G={\rm SO}(n)$, $K={\rm SO}(n-1))$ are the Gegenbauer or ultraspherical polynomials if one considers the irreducible representations of ${\rm SO}(n)$ ...
5
votes
0answers
163 views

Degree of Chebyshev polynomial necessary

In general, given $0<a<1$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[1-\frac{a}2,1+\frac{a}2]$ at every $x\in[1-a,1+a]$ and $f(0)=0$. What is minimum degree that is ...
0
votes
1answer
289 views

cup-length of the first Chern class of complex grassmannian

Let $G_2(\mathbb{C}^{n+1})$ be the complex grassmannian. Then the cohomology ring $H^*(G_2(\mathbb{C}^{n+1});\mathbb{C})=\mathbb{C}[c_1,c_2]/(f_n,f_{n+1})$, where ...
8
votes
3answers
472 views

Degree necessary of a polynomial?

Given $-1<a<b<0$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[a,b]$ at every $x\in[b^2,a^2]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that ...