Questions tagged [polynomials]

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, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.

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34 views

Reference for a real algebraic geometry problem

Disclaimer: I am not a mathematician by training. I encountered the following problem in my research. Assume that I have $N$ real variables $x_i$. I am given $N$ homogeneous polynomials in $x_i$ each ...
6
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1answer
188 views

Irreducibility of a polynomial when the sum of its coefficients is prime

I came up with the following proposition, but don't know how to prove it. I used Maple to see that it holds when $ a + b + c + d <300 $. Let $a,b,c$ and $d$ be non-negative integers such that $d\...
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93 views

Primes which do not divide certain homogeneous polynomials

It is known that if $x^2 + y^2 = z^2$ is a primitive Pythagorean triplet then $z$ is not divisible by any prime of the form $4k-1$. The following is a generalization of this classical result which ...
3
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110 views

Special subalgebras of $\mathbb{C}[t,x_1,\ldots, x_n]\partial_{x_1}\oplus\ldots\oplus\mathbb{C}[t,x_1,\ldots, x_n]\partial_{x_n}$

Consider the polynomial algebra $\mathbb{C}[t, x_1,\ldots, x_n]$. Denote the $\mathbb{C}[t]~-$ Lie algebra $$\mathcal{D}=\mathbb{C}[t, x_1,\ldots, x_n]\partial_{x_1}\oplus\ldots\oplus\mathbb{C}[t, x_1,...
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How to construct such a series of “partial” symmetric polynomials?

As we know, a symmetric polynomial is a polynomial P(X1, X2, …, Xn) in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, P is a symmetric ...
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1answer
139 views

Polynomials of minimum degree that interpolate primes in intervals

Given an interval $[a,b]$ what is the minimum degree of univariate polynomials in $\mathbb Q[x]$ that passes through all primes between $a$ and $b$ (denoted by $\mathbb P[a,b]$ with total number of ...
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72 views

Characteristic polynomial of the Gcd matrix

Let $A_n$ be the $n \times n$-matrix with entries $Gcd(i,j)$ and $f_n$ the characteristic polynomial of $A_n$. Question: Is $f_n$ irreducible for all $n$ except $n=8$? This is true for $n \leq 60$.
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71 views

A polynomial formed from the roots of another polynomial ad infinitum

Let $P(x)$ be a monic polynomial of degree $d$ with complex coefficients. Let $r_1(P),r_2(P),\dots, r_d(P)$ denote the set of roots, ordered so that $|r_1(P)| \leq |r_2(P)|\leq\dots\leq |r_d(P)|$. ...
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75 views

Concavity of a rational function

Let $n\in\mathbb{N}$ arbitrary but fixed. Consider the polynomial function $Q_n:\mathbb{R}\to\mathbb{R}$ given by $$ Q_n(x):=(x^2-1^2)^2(x^2-2^2)^2...(x^2-n^2)^2-1^42^4...n^4. $$ I would like to prove ...
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82 views

When does this limiting ratio give a real root $x$ to the equation of the form $\sum\limits_{k=0}^d \frac{x^k a_{k+1}}{k!}=0$?

By searching the Inverse Symbolic Calculator, we appear to be able to make the following conjecture about a real root to the equation: $$\sum\limits_{k=0}^d \frac{x^k a_{k+1}}{k!}=0 \tag{1}$$ Let the ...
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2answers
433 views

Approximation of smooth diffeomorphisms by polynomial diffeomorphisms?

Is it possible to (locally) approximate an arbitrary smooth diffeomorphism by a polynomial diffeomorphism? More precisely: Let $f:\mathbb{R}^d\rightarrow\mathbb{R}^d$ be a smooth diffeomorphism for $d&...
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1answer
83 views

Diophantine equations that involve Gregory coefficients: a computational exercise

In this post that I've asked three weeks ago with same title in Mathematics Stack Exchange and identificator 3692235, for integers $k\geq 1$, we denote the Gregory coefficients as $G_k$. Wikipedia has ...
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73 views

Class number of certain polynomials

Let $f_n(x)=x^n-\sum\limits_{i=0}^{n-1}{x^i}$ and $A_n$ the number field corresponding to $f_n$. Question: Is the class number of $A_n$ always equal to one, or equivalently, is the ring of integers ...
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46 views

The role of a combination of Eneström-Kakeya and Gauss-Lucas theorems: reference request or soft question, asking for this combination as tool

In past days I was trying to create problems or direct applications invoking Eneström—Kakeya and Gauss-Lucas theorems for certain arithmetic functions that I know from analytic number theory. These ...
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60 views

Sum of squares of polynomials in one variable with missing powers

As we known, a positive polynomial in $\mathbb{R}\left[x\right]$ can be expressed as a sum of squares of polynomials. The problem is that whether this holds if some powers is missed. Let $A$ be a ...
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63 views

Honda-Tate theorem and prescribing roots of $L$-functions

I'm currently working with Artin $L$-functions in function field extensions (on one variable, over finite fields). I have heard about a theorem of Honda-Tate which vaguely states that a polynomial $P \...
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1answer
230 views

Number of solutions of a degree 4 polynomial equation over a finite field

Suppose that $q$ is a prime power and $\xi, \eta\in \mathbb{F}_q$ are nonzero. A computer calculation for $q<70$ suggests that the number $N$ of $4$-tuples $(a,b,c,d)\in\mathbb{F}_q^{4}$ satisfying ...
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1answer
374 views

A simple proof that non negative polynomials have even degree [closed]

I am looking for a simple proof that a non-negative polynomial in n variables has always even degree. I have proved it but using Artin's Theorem ( Every non-negative polynomial is the sum of squares ...
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Bivariate basis functions with span that is closed under rotations in coordinate system

Consider a set of linearly independent functions $\{(x\in\mathbb{R},y\in\mathbb{R})\mapsto f_i(x,y)\in\mathbb{R}\}$ with the property that for any given $\theta\in\mathbb{R}$ and any given $\{a_i\in\...
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1answer
169 views

On a quadratic diophantine equation

Given a quadratic diophantine equation in $\mathbb Z[x,y,z]$ of form $$ax^2+by^2+cx+dy+ez+f=0$$ are there standard methods to solve for it when $$\|(x^2,y^2,z)\|_\infty\leq e^{1/2}$$ $$\|(a,b,c,d,e,f)...
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1answer
120 views

How to do a multinomial theorem sum faster

For example we have this question : Find the coefficient of $x^6$ in the following $\frac{\left(x^{2}+x+2\right)^{9}}{20}$ So using multinomial Theorem which is this : $\left(x_{1}+x_{2}+\cdots+x_{...
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141 views

On the $\mathsf{LCM}$ of a set of integers defined by moduli of powers

For integers $a,b,t$ define $$\mathcal R_t(a,b)=\{q\in\mathbb Z\cap[1,\min(a^t,b^t)]: a^t\equiv b^t\bmod q\}$$ and $\mathsf{LCM}(\mathcal R_t(a,b))$ to be $\mathsf{LCM}$ of all entries in $\mathcal ...
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Interpolator coefficients using collocation method for Radau IIA method (in solve_ivp)

I am trying to figure out how to calculate the following coefficients in the source code for the Radau integration method. The source code is here. I read that it uses a cubic polynomial to ...
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1answer
84 views

Roots for $p(w)=n+\sum_{j=1}^{m}\frac{v_{j}}{w-v_{j}}$

Let $v_{j}\in \mathbb{C}, 1\leq j\leq m$ and $w\in \mathbb{C}\setminus \{v_{j}\}_{j=1}^{m}$ and $n>0$. Q: Can we say anything about the m roots $w_{1},...,w_{m}$ of $$p(w)=n+\sum_{j=1}^{m}\frac{...
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161 views

No common roots of complex polynomial and of its derivative

Our specific context Here is our specific contour integral $$\int_{\Gamma_{0}}F\big(\sum_{w:p_{z}(w)=0}\frac{1}{w^{a}}\frac{1}{n+\sum_{j=1}^{m}\frac{v_{j}}{w-v_{j}}} \big)\frac{dz}{z},$$ ...
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1answer
205 views

Coefficients of $(2+x+x^2)^n$ from trinomial coefficients

I would like to be able to express the coefficients of $(2+x+x^2)^n$ in terms of the trinomial coefficients studied by Euler, ${n \choose \ell}_2 = [x^\ell](1+x+x^2)^n$ where $[x^\ell]$ denotes the ...
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435 views

Is there a notion of polynomial ring in “one half variable”?

Let $C$ be the category of commutative rings. Is there a functor $F :C \to C$ such that $F(F(R)) \cong R[X]$ for every commutative ring $R$ ? (Here, we may assume those isomorphisms to be natural ...
2
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1answer
68 views

Existence of solutions of polynomials systems (and their “rough” shape) over $\mathbb{R}$ & friends with positive-dimensional ideals

This is a follow-up (but self-contained) question to my previous one. There I asked about state-of-the-art methods to solve multivariate polynomials systems over non-algebraically closed fields in ...
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1answer
1k views

SOS polynomials with integer coefficients

A well known theorem of Polya and Szego says that every non-negative univariate polynomial $p(x)$ can be expressed as the sum of exactly two squares: $p(x) = (f(x))^2 + (g(x))^2$ for some $f, g$. ...
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2answers
473 views

What is the state-of-the-art for solving polynomials systems over fields that are not algebraically closed?

I am not working in the field of algorithmic algebraic geometry - yet, for my current work, I need some results from it. More specifically, what is the state-of-the-art when it comes to solving (...
4
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1answer
454 views

A problem on polynomials

Let $P(z)$ be a polynomial of degree $n$ with $|P(z)|\leq 1$ on $|z|=1$ and $P_m(z)$ be a partial sum of $P(z).$ How large $P_m(z)$ can be on $|z|=1?$
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146 views

Question about polynomials in $\mathbb{C}[x, y, z]$

What can be said about pairs of polynomials $P, Q\in\mathbb{C}[x, y, z]$, such that $\frac{\partial P}{\partial y}\frac{\partial Q}{\partial z} - \frac{\partial P}{\partial z}\frac{\partial Q}{\...
2
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0answers
104 views

Reducible polynomial among sequence of polynomials

Let $a_1$ and $a_2$ be two elements of a finte field $\mathbb{F}_{2^m}$ of even characteristic and $a_1^2\neq a_2$. Is it true that there always exists an element $a\in\{a_1,a_2,a_3,\ldots,a_{2^m}|a_{...
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2answers
330 views

The first part of the Hilbert sixteenth problem for elliptic polynomials

A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its highest homogeneous part does not vanish on $\mathbb{R}^2\setminus\{0\}$. Inspired by the first part of the Hilbert ...
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79 views

Chinese remaindering to solve solvable diophantine equations

Given a diophantine equation $$f(x_1,\dots,x_z)=0$$ where $f(x_1,\dots,x_z)\in\mathbb Z[x_1,\dots,x_z]$ is of total degree $d$ and each variable degree $d_i$ where $i\in\{1,\dots,z\}$ there is no ...
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83 views

Meaning and/or source of polynomials residually of the form $x^n(x-1)$ in Gabber's characterization of Henselian pairs?

Lemma 09XI in the stacks project includes a characterization (#5) by Gabber of Henselian pairs $(A,I)$: first, $I\leq \mathrm J(A)$ is contained in the Jacobson radical and second, every monic $f\in ...
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27 views

Fixing coefficients using analytic structure

I am trying to understand an exercise in a set of lecture notes on random matrices - Eynard - Random matrices - given in the paragraph following (4.6.60) (pp. 70–71). Specifically, I am trying to fix ...
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0answers
91 views

Phase angles of a complex eigenvector

I have the following system for $\lambda \in \Bbb C, \lambda \neq 0$ and $\pmb{p},\pmb{q} \in \Bbb C^n$, $(\pmb{p}^T, \pmb{q}^T)\neq0$: $$\begin{cases} F(\lambda) \pmb{p} - g(\lambda) \pmb{q} - \...
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206 views

Galois groups of special polynomials

This question is motivated by long experiments with GAP. Call a monic polynomial with integer coefficients special in case it is irreducible and has only coefficients $-1$, $0$ or $1$. Let $n \geq 5$....
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1answer
100 views

A question on multiplicity of complex polynomial [closed]

This is not a research level question. But due to some reason I can't ask this question on Math Stack Exchange. So, I am asking this question here. By definition we know that we can measure the ...
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169 views

Galois group of polynomials related to Fibonacci and Catalan numbers

Let $F_n$ be the Fibonacci and $C_n$ the Catalan numbers. Define a polynomial by $F_n(x):=\sum\limits_{k=1}^{n}{F_k x^{n-k}}$. For example $F_8(x)=x^7+x^6+2x^5+3x^4+5x^3+8x^2+13x+21$. And another ...
3
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1answer
208 views

Polynomial satisfying a functional equation [closed]

I am currently stuck with the following question: Let $q$ be a polynomial of degree $n+1$ with distinct positive zeros $x_0, ... , x_n$. Find a polynomial $p \in P_n$ that satisfies the functional ...
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0answers
47 views

On a structural decomposition of polynomials based on integral roots

Given an irreducible polynomial of structure $$f(x,y)=\sum_{\substack{i,j\in\{0,1,2\}\\i+j\leq3}}a_{i, j}x^iy^j\in\mathbb Z[x,y]$$ with $a_{2,1}a_{1,2}a_{1,1}a_{1,0}a_{0,1}a_{0,0}\neq0$ if $f(m,n)=0$ ...
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0answers
45 views

Is there a classification of higher-degree generalisations of confocal conic sections?

The 1-parameter families of ellipses and hyperbolas with a given pair of points in the plane as their foci yield “orthogonal double-foliations” of the plane. That is, once the foci are specified, any ...
2
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66 views

Affine projection of polynomials for a given set of points

(Not sure this question fits here, I will remove it in case it doesn't) Let $F_{\mathrm{ML}}[x_1, \ldots,x_n]$ denote the set of multilinear polynomials over a finite field $F=\mathbb{F}_q$ (i.e. a ...
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0answers
241 views

Can a polynomial family of curves be identified by a polynomial satisfied by its derivatives?

Suppose we have a 1-parameter family of curves that foliate a subset of the plane, defined by a function that is a polynomial in the coordinates $(x,y)$ and a parameter that identifies the individual ...
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175 views

is it possible to prove this :$\sum_{n\le X}\tan(n!)=\mathcal{O}(X^{\alpha})$?, $\alpha <1$?

This series $\sum_{n\geq 0}\tan(n!)$ probably diverges. I want to find a polynomial bounding $|\sum_{n\le X}\tan(n!)|$. Is it possible to prove that $\sum_{n\le X}\tan(n!)=\mathcal{O}(X^{\alpha})$ ...
2
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1answer
82 views

When is a set defined by multivariate polynomial inequalities convex?

Consider the set of real numbers given by $$S = \{(a,b,c,d,e,f,g,h) \in [0,1]^8 : 0 \le \frac{e(g-h)}{b(g-f)} \le 1 \text{ and } 0 \le \frac{e(h-f)}{(1-b)(g-f)} \le 1\}$$ Note that this set can also ...
0
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1answer
118 views

Matrix whose entries are given by polynomial $A_{ij} = p(\lambda_i, \lambda_j)$; when is it positive semidefinite?

Let $A$ be a matrix whose entries are given by a polynomial, $$ A_{ij} = p(\lambda_i, \lambda_j) $$ where $p(\lambda_i,\lambda_j) = p(\lambda_j,\lambda_i)$ is symmetric. Are there standard methods ...
7
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0answers
411 views

Again, polynomial bijection $f:{\mathbb Q}\times{\mathbb Q}\times{\mathbb Q}\to {\mathbb Q}$

Assume that there is no polynomial $f(x,y)\in{\mathbb Q}[x,y]{}$ such that $f\colon {\mathbb Q}\times{\mathbb Q}\to {\mathbb Q}$ is a bjiection. Does this imply that there is no polynomial $f(x,y,z)\...

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