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.
2,555
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Partition the rationals with respect to a multivariate polynomial which sends classes to classes
Let $R$ be a commutative ring and let $f\in R[x_1,x_2,\cdots,x_{n-1}],n\geq 2$ be a polynomial.
Definition: We say $f$ is $n$-severable over $R$ if there exists a partition (of set) $$R=\coprod_{i=...
- 171
7
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0
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232
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Unprovable integer identity involving exponentiation
I was thinking about Tarski's problem, and was wondering what happens if we have a theory $T$ with two sorts $N,Z$ with intended interpretations $\def\nn{\mathbb{N}}$$\def\zz{\mathbb{Z}}$$\nn,\zz$ ...
- 2,733
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181
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Resultant of two special trinomials
Consider $f(x)=x^n-x^s-1$ and $g(x)=x^i-x^j-1$ , I want to find $Resultant(f,g)$. It is well known that it is determinant of a Sylvester matrix but, I am finding it to obscure to evaluate in that way. ...
- 306
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0
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357
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Intersections of ideals in polynomial rings with countably many variables
Fix a field $k$ and let $R = k[x_1,x_2,\ldots]$. Say that an ideal $I \subset R$ is generated in finite degree if there exists a generating set $S$ for $I$ (possibly infinite) and an integer $n$ such ...
- 71
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211
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Increasing derivatives of recursively defined polynomials
Consider recursively defined polynomials $f_0(x)=x$ and $f_{n+1}(x)=f_n(x)−f_n'(x) x (1−x)$.
These polynomials have some special properties, for example $f_n(0)=0$, $f_n(1)=1$, and all $n+1$ roots of ...
- 225
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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 ...
7
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0
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227
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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 ...
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250
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How many simultaneous polynomial equations of degree 2 can software solve today?
Consider the following problem:
Input: $n$ polynomial equations of degree $2$ in approximately $n$ variables.
Each equation contains about $\sqrt{n}$ monomials.
We would like to find one simultaneous ...
- 71
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547
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Factors of the polynomial $X^n-a$
I am interested in the polynomial $X^n-a$ in $\mathbb{Q}[X]$, for some $a\in \mathbb{Q}^*$, and would like to know the irreducible factors of it. Is there something in the literature which gives a ...
- 7,652
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337
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Polynomials and divided differences
I would greatly appreciate any hint for proving the following.
Question: Let $f:[0, 1] \to {\bf R}$. Can it be proved that if $[0, 1/(N+m),\dots, (N+m)/(N+m) ; f ]=0$ for all $m=1,2, 3,\dots$, then $...
- 71
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504
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$a^5+b^5=c^5+d^5$ and polynomial identities
No nontrivial integer solutions to $$ a^5+b^5=c^5+d^5 \qquad (1)$$ are known.
(1) has infinitely many solutions in an extension of $\mathbb{Z}$
(root of $9-15x+37x^2 $ ) resulting
from a genus 0 ...
- 24.2k
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474
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How many values a polynomial map misses?
Let $F$ be a field. For a uni-variate polynomial $f(x)$ over $F$,let $M_f(F)$ denote the number of values that $f$ misses, that is, the cardinality of the subset $F - f(F)$ in $F$. Assume that $f$ is ...
- 71
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486
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"Consecutive" irreducible polynomials
If $P\in {\mathbb Z}[X]$ is a polynomial of degree $2$, then
it is easy to see that for any integer $m$, at least one of the polynomials
$P-(m+1),P-(m+2),P-(m+3),P-(m+4)$ is irreducible in ${\mathbb Z}...
- 3,565
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When is the Locus of Equi-modular points of two monic polynomials with integer coefficients contained in the unit disk?
If $\lambda_{1}(z)$ and $\lambda_{2}(z)$ are two monic polynomials (relatively prime) with integer coefficients and $$\Gamma:=\lbrace z \rm{\ s.t.\ } |\lambda_{1}(z)|=|\lambda_{2}(z)|\rbrace,$$ when ...
- 103
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roots of higher derivatives of exponential
Consider the Gaussian function $f(z)=e^{-z^2}$ which has no zeros on the complex domain. Let $D$ denote derivative w.r.t. the variable $z$.
Question. Is it true that $D^nf(z)=0$ has only real roots ...
- 41.8k
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2
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788
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Is every square root of an integer a linear combination of cosines of $\pi$-rational angles?
For example, $\sqrt 2 = 2 \cos (\pi/4)$, $\sqrt 3 = 2 \cos(\pi/6)$, and $\sqrt 5 = 4 \cos(\pi/5) + 1$. Is it true that any integer's square root can be expressed as a (rational) linear combinations of ...
- 1,461
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2
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923
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Counterexample to Proposition of Granville related to abc conjecture
Looks like there is counterexample to Proposition related to
abc conjecture. Confusion is likely.
From RATIONAL AND INTEGRAL POINTS ON QUADRATIC TWISTS OF A GIVEN HYPERELLIPTIC CURVE, Andrew ...
- 24.2k
6
votes
4
answers
663
views
What is the essence of the constant factor in the standard definitions of the discriminant?
Let $f(x) = x^m+\sum_{j=0}^{m-1}f_{m-j}x^j\in P[x]$ be a monic polynomial over a field $P$ and let $f(x) = (x-\alpha_1)\cdot\ldots\cdot(x-\alpha_m)$ be a factorization of $f$ over an extension field $...
6
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2
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Integer roots of a symmetric polynomial
The question is very simple and I apologize for that, but I am not an expert of this kind of problem.
Given the polynomial
$$ P(x_1,\ldots,x_{2n})=x_1^2+\ldots+x_n^2-x_{n+1}^2-\ldots-x_{2n}^2,$$
I ...
- 403
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2
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Polynomial reducible modulo every integer
Hi,
let $f\in\mathbb{Z}[X]$ be a monic polynomial. Assume that the reduction of $f$ modulo $m $ is reducible for all integers $m\geq 2$.
Q1: Is $f$ reducible in $\mathbb{Z}[X]$ ?
I've thought ...
- 69
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1
answer
1k
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If some powers of polynomials are linearly independent, does it imply higher powers are also independent?
Let $P_1,\dotsc,P_k$ be polynomials. Assume they are pairwise non-proportional (i.e., any two of them are linearly independent). Suppose $N$ is a power such that $P_1^N,\dotsc,P_k^N$ are linearly ...
- 6,197
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Irreducible polynomial $p_{n}(x)=\sum_{k=0}^{n}\frac{x^k}{k!}$ for all positive integers $n$
Let $n$ be a positive integer greater than $1$, and define the polynomial $$p_{n}(x)=\sum_{k=0}^{n}\dfrac{x^k}{k!}$$
Is $p_{n}(x)$ irreducible in $\mathbf{Q}[x]$?
I can show it when $n$ is a ...
- 4,230
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votes
4
answers
15k
views
Number of polynomial terms for certain degree and certain number of variables [closed]
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 ...
- 179
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2
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988
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Irreducibility for multivariate polynomial polynomial generated from sum of irreducible polynomial in one variable
Could any tell me if a multivariate polynomial generated from the sum of irreducible single variable polynomial is irreducible?
For example, f(x)=x^2+2x+2, g(x)=x^2+3x+3, h(x)=x^3+2x^2+2x+2 all of ...
- 63
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2
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2k
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A family of polynomials with symmetric galois group
Consider the following family of polynomials in $K[x,y]$, where $K$ has characteristic zero:
$f_n(x,y)=(x+y)^n+(x-1)y^n,$
for $n\geq 3$. I can prove that $f_n(x,y)$ has an irreducible factor of ...
- 399
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4
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When does a polynomial have all pure imaginary roots? [closed]
Let $P(x)=x^{n}+a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}$, where $a_1, a_2, \dots, a_n$ are intergers.
Question 1. When does the polynomial $P(x)$ have its zeros all being pure imaginary or zero(here 0 is ...
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722
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Deceptive linear algebra problem
Does anyone recognize this problem? There are $2p$ equations and $2p$ unknowns, and it feels like a classic, but I've never encountered it before:
Given $d_1, d_2, \ldots d_p$, find $a_1, a_2, \ldots ...
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How to show simply that $e^{\frac{x}{2}}\int^\infty_0 e^{-t}t^{n-\frac{1}{2}}\cos(2\sqrt{xt})dt=O\big(\frac{n!}{\sqrt{n}}\big)$?
Can we prove, without using Laguerre polynomials, that
$f_n(x)=O(\frac{n!}{\sqrt{n}})$ i.e. that
$$
\exists C>0, \exists N\in\mathbb N, \forall x\geq0, \forall n\geq N :\ \big| f_n(x) \big|\leq C \...
- 1,503
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722
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Multivariate polynomial is never squarefree at integers
Let $f(x_1,\dots,x_n)$ be squarefree polynomial with integer
coefficients. Assume $f$ at integers is not always divisible
by a fixed square $m^2 > 1$.
Is it possible $f$ to never be squarefree at ...
- 24.2k
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2
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2k
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When does a rational function have infinitely many integer values for integer inputs?
Consider rational functions $F(x)=P(x)/Q(x)$ with $P(x),Q(x) \in \mathbb{Z}[x]$. I'd like to know when I can expect $F(k) \in \mathbb{Z}$ for infinitely many positive integers $k$. Of course this ...
- 1,309
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answers
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How to solve Diophantine equations in $F_{p}$?
For example, how to solve the equation $\sum^{p-1}_{i}x_{i}^{2}=0$ in $F_{p}$? This is not a homework problem. I think it should have a definite answer, so not an open problem. I just don't know how ...
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Asymptotic series for roots of polynomials
Let $f(z) = z + z^2 + z^3$. Then for large $n$, $f(z) = n$ has a real solution near $n^{1/3}$, which we call $r(n)$. This appears to have an asymptotic series in descending powers of $n^{1/3}$, ...
- 13.9k
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Prove that $\Bbb C[x,y]/(x^3+y^3-1)$ is not a UFD
I am posting this question on MO since I haven't received any answers on MSE.
Below is my (very elementary) attempt. Feel free to post a solution using facts in algebraic geometry and facts about ...
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Is this a semi algebraic set?
Put $$A=\{(a_{0},a_{1},\ldots,a_{n}) \in \mathbb{C}^{n+1}\mid p(z)=a_{0}+a_{1}z+\ldots a_{n}z^{n} \;\;\text{is a one-to one function on the unit disc} \{z\in \mathbb{C} \mid |z|\leq 1\}$$
Is $\{(|...
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Seeking an explanation for a peculiar factorization
Recently during my work, I encountered the following family of sextic polynomials
$$\displaystyle x^6 - 3x^5 + cx^4 + (5 - 2c)x^3 + cx^2 - 3x + 1.$$
By plugging in various values of $c$, I noticed ...
- 25.5k
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5
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646
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Is there a name for the involution on Laurent polynomials?
This is a simple terminology question: I want to know if the involution $z \mapsto z^{-1}$ on Laurent polynomials (over some ring, I happen to be working over $\mathbb{Z}$ but that's not important) ...
6
votes
1
answer
514
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Construction of a symmetric polynomial in the roots that acts like the discriminant
The discriminant $\Delta(P)$ of a monic polynomial $P(x)=x^n + a_{n-1} x^{n-1} + \dotsb + a_0$ of degree $n$, when expanded (using elementary symmetric polynomials), is a symmetric polynomial of ...
6
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3
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490
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Uniformly approximating a function and its derivative using polynomials
I'm struggling either proving or disproving the following statement:
Let $K\subset \mathbb{R}$ be compact, and $S = \mathrm{span}\{p_k, k = 0, 1, \ldots\}$, where $p_k$'s are polynomials over $K$. If ...
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votes
1
answer
956
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Solution to sixth order equation
I'm dealing with the expression $x = \frac{1}{3}y(y+1)(2y+1)^2(2y^2+2y+1)$. What is this approximately, if one is explicitly writing y in terms of x? There's no general formula for sixth powers ...
6
votes
1
answer
708
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Find the maximum of $|a_{p}|$, if $a_0+a_1x+\dots+a_nx^n:[-1,1]\mapsto [-1,1]$
Let $n$ be a given positive integer, and let $f(x)=\displaystyle\sum_{k=0}^{n}a_{k}x^k$, where $a_{i}\in \mathbb{R}$, $0 \le i \le n$. If
$$|f(x)|\le 1,\qquad \text{for } ~|x|\le 1,$$
what is the ...
- 4,230
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votes
1
answer
813
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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 $O(n!)$-...
- 61
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3
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490
views
Polynomial threading through a monotone corridor
I have a need to find a polynomial of minimal degree that connects
two points and stays within a given
"corridor," by which I mean an $x$-monotone polygon.
Here is an example:
&...
- 149k
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votes
4
answers
908
views
Can this nested sum be expressed in terms of generalized harmonic numbers and the cycle index polynomials of the symmetric groups?
For a paper I was working on recently I needed to find the value of the following sum:
$$S(n,k) = \sum_{i_1 = 1}^n \sum_{i_2 = i_1+1}^n \cdots \sum_{i_k=i_{k-1}+1}^n \frac{1}{i_1 i_2 \cdots i_k}.$$
...
- 3,253
6
votes
2
answers
462
views
Optimal polynomial approximation of rational function $\frac{1}{1-x}$
I've been working on the following polynomial approximation problem. I want to find the optimal Chebyshev approximation of the rational function $\frac{1}{1-x}$ on the real interval $x\in[-\rho, \rho]$...
- 63
6
votes
2
answers
663
views
Integral domain over which any non-constant, one variable, irreducible polynomial has degree 1
Let $R$ be an integral domain such that every non-constant, irreducible polynomial $f(X) \in R[X]$ has degree $1$.
Q. is it true that $R$ is a field?
If $0 \ne a \in R$ , then $X^2-a$ is ...
user111524
6
votes
2
answers
639
views
how to show the existence of root for a system of polynomial equations?
I come across a system of polynomial equations. Although I can solve it by Mathematica (numerically), I wonder whether it is possible to prove this root indeed exists. Any general method to do it?
For ...
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votes
2
answers
469
views
Why are most coefficients of these minimal polynomials divisible by $p$?
For an odd prime $p$, let $\zeta:=e^{\frac{\pi i}p}$ and choose odd $1<n<p$. Further let $q(x)$ and $r(x)$ be integer polynomials such that $r(x)$ has no common factor with $x^n+1$, and $\xi$ ...
- 13.2k
6
votes
3
answers
291
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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. ...
- 4,006
6
votes
2
answers
2k
views
Complexity of finding a rational root of a polynomial
This is inspired by this question. Let $f(x)=a_nx^n+...+a_0$ be a polynomial with rational coefficients. The sandard procedure of finding a rational root $p/q$ involves checking all $p$ that divide $...
user6976
6
votes
1
answer
885
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Resultant of linear combinations of Chebyshev polynomials of the second kind
The Chebyshev polynomial $U_n(x)$ of the second kind is characterized by
$$
U_n(\cos\theta)=\frac{\sin(n+1)\theta}{\sin(\theta)}.
$$
It seems that
$$\operatorname*{Res}_x \left( U_n(x)+tU_{n-1}(x),\...
- 437