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.
2,543
questions
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54
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Factorization of the symmetric function identity $E(t)=1/H(t)$ with the refined Euler characteristic polynomials of associahedra / Lagrange inversion
I've come across two matrix identities, flagged with daggers below, relating the two sets of elementary and complete homogeneous symmetric polynomials/functions via the two sets of refined Lah and ...
0
votes
1
answer
61
views
Can non-periodic discrete auto-correlation be inversed?
I'm trying to understand whether discrete auto-correlation can be reversed.
That is, we are given $t_0, \dots, t_n \in \mathbb C$ and a set of equations
$$
t_{k} = \sum\limits_{i=0}^{n-k} b_i b_{i+k},
...
36
votes
6
answers
3k
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Number of real roots of 0,1 polynomial
$0,1$ polynomial has coefficients from $\{0,1\}$.
I investigate the number of roots in such polynomials.
We are talking about real roots, and multiples are counted only once.
It was found numerically ...
-5
votes
1
answer
69
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Application of Resultant in Computer Algebra [closed]
Can you guys give me some application of resultant in Computer Algebra, it will be amazing if you guys can give me some paper or book to read more. Thanks so much
3
votes
1
answer
130
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Homogeneous regular (= polynomial component) maps with odd degree and their being global homeomorphisms in dimensions higher than one?
Let $F:\mathbb{R}^m \to\mathbb{R}^m, F:=(F_1\dots F_m)$ be a regular map, i.e. with components $F_i$ that are polynomials.
Assume further that each $F_i$ is an odd degree (say $d$) homogenous ...
0
votes
0
answers
71
views
On the multiplicative group of quotients of polynomial rings
Related to this.
The $p+1$ factorization algorithm works over $\mathbb{Z}/n\mathbb{Z}[x]/f(x)$
and hopes $p+1$ to be smooth.
We are trying to generalize this to multivariate case
and also try to find ...
3
votes
2
answers
302
views
Where to find or how to prove that the ratio of two Bernoulli polynomials is increasing?
It is well known that the classical Bernoulli polynomials $B_j(t)$ are generated by
\begin{equation*}
\frac{s\operatorname{e}^{ts}}{\operatorname{e}^s-1}=\sum_{j=0}^{\infty}B_j(t)\frac{s^j}{j!}, \quad ...
3
votes
1
answer
75
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Evaluating the generalized continued fraction obtained from the factorization of a bivariate polynomial equation
Happy New Year, MO community!
We need someone expert in Generalized Continued Fractions (GCFs), with a deep knowledge of the GCFs’ convergence properties, to solve the following problem.
PROBLEM
...
2
votes
0
answers
129
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Parametrizing "ternary cubic equals a square"
I am interested in an equation of the form
$$\displaystyle y^2 = f(x_1, x_2, x_3),$$
where $f \in \mathbb{Z}[x_1, x_2, x_3]$ is a ternary cubic form. In particular, I am looking for an analogue of the ...
2
votes
0
answers
146
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Large prime divisors of values of a polynomial, in a given residue class
Let $f(X) \in \mathbb{Z}[X]$ be an irreducible polynomial of degree $d \geq 2$. Let $q \in \mathbb{N}$ be an integer, and let $q \mathbb{Z} + r$ be a residue class that contains infinitely many primes ...
0
votes
0
answers
120
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Nilpotent elements of $(\mathbb{Z}/n \mathbb{Z})[x_1,...,x_m]/\langle f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)\rangle$
This is generalization of the univariate case
and also related to open problem.
Let $n,k,m,B>1$ be positive integers and $f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)$ be polynomials with ...
4
votes
0
answers
166
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Nilpotent elements of $K=(\mathbb{Z}/n \mathbb{Z})[x]/f(x)$
This is related to an open problem.
Let $n$ be integer and $f(x)$ polynomial with integer coefficients and set $K=(\mathbb{Z}/n \mathbb{Z})[x]/f(x)$.
Let $S$ be the set of degree 2 nilpotent elements ...
2
votes
1
answer
257
views
Curious sequences of polynomials
Given an integer $k\geq 2$, and $k+1$ invertible initial
values $s_0,s_1,\ldots,s_k$ in some commutative ring $\mathcal A$
we set
$$s_{n+1}=\frac{\sum_{j=1}^ks_{n+1-j}^2+q \sum_{j=1}^{k-1}s_{n+1-j}s_{...
1
vote
0
answers
220
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Is $\sum\limits_{k=1}^nk^m=S_3(n)\cdot\dfrac{P_{m-3}(n)}{N_m}$ for odd $m>1,\sum\limits_{k=1}^nk^m=S_2(n)\cdot\dfrac{P_{m-2}'(n)}{N_m}$ for even $m$?
I asked this question here
When I was in high school, I was fascinated by
$$
\sum\limits_{k=1}^n k= \frac{n(n+1)}{2}
$$ so I tried to find the general value of the sum
$$
\sum\limits_{k=1}^n k^m\;\...
1
vote
0
answers
270
views
On fifth powers forming a Sidon set
We call a set of natural numbers $\mathcal S$ to be a Sidon Set if $a+b=c+d$ for $a,b,c,d\in \mathcal S$ implies $\{a,b\}=\{c,d\}$. In other words, all pairwise sums are distinct.
Erdős conjectured ...
2
votes
1
answer
127
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Are there a few input bits that randomize the output of an $\mathbb{F}_2$ polynomial?
Suppose $f:\mathbb{F}_2^n\rightarrow \mathbb{F}_2$ is a degree $d$ polynomial and $\epsilon>0$ is some real number. Does there necessarily exist a set $C\subset [n]$ of coordinates with the size of ...
2
votes
2
answers
311
views
Uniqueness of sum of squares representation
Given a polynomial $f(x) \in \mathbb{R}[x] = \mathbb{R}[x_{1},\dots,x_{n}]$. We say $f(x)$ is sum of squares(SOS) if there are polynomials, $p_{1},\dots,p_{k}$ such that $f = p_{1}^{2} + \dots+p_{k}^{...
4
votes
1
answer
115
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How are Lie groups and polynomial resolvents related?
I came across the following sentence in Stevenhagen and Lenstra's wonderful little article Chebotarëv and his density theorem:
Nikolai's interest in [polynomial] resolvents led him to study Lie ...
7
votes
1
answer
295
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Approximating functions on the real line
While it is not possible to approximate any function with polynomials on the entire real line, I am wondering if there are modified conditions under which the approximation is possible. Consider $f \...
1
vote
1
answer
76
views
Surprising numerical coincidence while interpolating on Smolyak grid
I was plotting 2-D shape functions for linear interpolation on a Smolyak sparse grid of level 2 associated to Gauss-Lobatto-Chebyshev nodes(cf https://en.wikipedia.org/wiki/Sparse_grid ), when I came ...
2
votes
0
answers
78
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Closed form solutions to polynomial operator equations
To the best of my knowledge the problem at hand is a generalisation of monic matrix polynomials. Can a closed form solution to the following equation be found,
$$u_3A_3X^3B_3 + u_2A_2X^2B_2 + ...
1
vote
0
answers
147
views
Geometric construction of real root of quintic using marked ruler and compass
My question is motivated by a geometry problem about special folded rectangle:
'A rectangle with sides a, b (a<b) is folded along the line that passes through the center of the rectangle in order ...
8
votes
0
answers
488
views
An algebraic version of the implicit function theorem for integers
$ \def \x {\boldsymbol x}
\def \a {\boldsymbol a}
\def \Z {\mathbb Z} $
The famous version of the implicit function theorem (IFT) starts with the assumption of continuous differentiability on the ...
0
votes
0
answers
57
views
The discrete orthogonal polynomials
I want a document or something that explains the following proposition:
The discrete orthogonal polynomials are the polynomial solutions of the given diference equation:
$$
\sigma(x)\Delta\nabla P_n(...
1
vote
1
answer
81
views
Formulas for partial composed product
Let $A(x) = \prod\limits_i (x-\lambda_i)$ and $B(x) = \prod\limits_j (x-\mu_j)$. Then, their composed product is defined as
$$
(A*B)(x) = \prod\limits_{i,j} (x-\lambda_i \mu_j).
$$
Generally, we can ...
2
votes
1
answer
187
views
Slicing bivariate exponential generating functions on x and y
Let $F(x, y) = e^{y D(x)}$ be a generating function for sets of objects enumerated by $D(x)$ that also keeps track of the number of sets (enumerated by the variable $y$, while $x$ enumerates the total ...
2
votes
0
answers
60
views
Iterated chaos expansion
Using the notations from Normal Approximations with Malliavin Calculus, Chapter 2
random variables $F$ in the probability space generated by an iso-normal Gaussian family $X(h)$,
$$E[X(h)X(g)] = \...
16
votes
1
answer
750
views
Find a special integer coefficients polynomial which takes small absolute value on [0,4]
The question is easy to state: Is there a non-constant $f\in\mathbb{Z}[x]$ such that for all $x\in [0,4]$, we have $|f(x)|\leq 1$? I do not know where to find a useful reference for it.
I did a few ...
0
votes
1
answer
108
views
Sufficient conditions for ensuring that a monic polynomial in $\mathbf{Z}[x]$ possesses exclusively simple roots
I am seeking sufficient conditions to ensure that a monic polynomial, denoted as $f$ in $\mathbf{Z}[x]$, possesses exclusively simple roots.
Based on an old paper (this reference), it has been ...
1
vote
0
answers
55
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Eigenvalues of a subset of matrix semigroup
My apologies for slightly longer post but I wanted to explain lower dimensional cases and their proofs before asking the actual question, which starts after the phrase The general case below.
A two-...
6
votes
2
answers
1k
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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 ...
17
votes
1
answer
660
views
Multiply an integer polynomial with another integer polynomial to get a "big" coefficient
I have copied this question from StackExchange, in the hope that some experts here can provide some relevant insight. Thanks to Greg Martin for improving the question.
Given $f(x) = a_0 + a_1 x + a_2 ...
6
votes
2
answers
606
views
Can this system of equations about Newton's formula have concrete result?
Try to solve this system of equations:
$$
S_1=x_1+\dots+x_n=a;\\
S_2=x_1^2+\dots+x_n^2=a;\\
{}\cdots\\
S_n=x_1^n+\dots+x_n^n=a;
$$
And find the value of $S_{n+1}=x_1^{n+1}+\dots+x_n^{n+1},a\in\mathbb{...
0
votes
0
answers
34
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polynomials defined on a non-uniform and uniform lattice
After reading about the definition of orthogonal dual Hahn polynomials on wiki, you will find the link below. I didn't understand the following sentence :
"In mathematics, the dual Hahn ...
2
votes
0
answers
124
views
Asymptotics of a "non-constant order" quadratic recurrence relation in two variables
Consider the following recurrence relation defined for two integer variables $H,n \geq 0$:
\begin{equation}
\gamma(H,n) = \sum_{K=0}^{\lfloor H/2 \rfloor} \gamma(K,n-1) \gamma(H-K,n-1)
\end{equation}
...
2
votes
2
answers
148
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On the number of values with exactly $k$ prime factors of a given polynomial
This is surely be a well studied problem. Let $f(x) \in \mathbb{Z}[x]$. Is there some $k \in \mathbb{N}$ such that there are infinitely many $n \in \mathbb{Z}$ where $f(n)$ has exactly $k$ prime ...
5
votes
1
answer
421
views
Does coefficient-wise limit preserve real-rootedness?
Let $P_n$, $n=1,2,\ldots$ be polynomials with real roots only (and real coefficients), and $P_n$ converge to a non-zero polynomial $Q$ coefficient-wise. Does it follow that $Q$ has real roots only?
...
9
votes
1
answer
650
views
Sequence of real-rooted polynomials
I've been interested in proving a log-concavity result via proving that certain family of polynomials is real-rooted. By performing a sequence of transformations, I can reduce that problem to proving ...
2
votes
2
answers
196
views
An identity for the ratio of two partial Bell polynomials
Let $B_{\ell,m}(x_1,x_2,\dotsc,x_{\ell-m+1})$ denote the Bell polynomials of the second kind (or say, partial Bell polynomials, (exponential) partial Bell partition polynomials). I knew that
the ...
4
votes
1
answer
239
views
Irreducible integral polynomials having roots module primes in arithmetic progressions
Let $f(x)$ be an irreducible polynomial with integer coefficients. One can show (see Exercise 7.2 in this paper of Lenstra) that if $f(x)=0$ has a solution mod $p$ for all but finitely many primes $p$...
0
votes
1
answer
185
views
Trying to solve for the remainder of $a^q$ modulo $q$
Let $q$ be a prime and $a$ be a number from $0$ to $q-1$ (not an equivalence class).
The elements $a^q$ are exactly the elements of order $q-1$ modulo $q^2$.
I'm trying to solve the equation:
$$a+2*\...
8
votes
1
answer
599
views
Connеcted components of irreducible algebraic varieties
I am wondering what is the possible (or maximum) number of connected components for an irreducible algebraic variety in $\mathbb R^n$ defined by a degree $d$ polynomial (i.e. hypersurface) in $\mathbb ...
1
vote
0
answers
108
views
Degrees of trigonometric numbers
For a rational number $r\in(0,1)$ the number $z=\sin(r\pi)$ is an algebraic number — such numbers appear to be called trigonometric numbers.
What is its degree?
That is, what is the minimal degree of ...
7
votes
1
answer
878
views
Can a field have an irreducible polynomial of any degree?
We all know that all the irreducible polynomials in $\mathbb{C}[x]$ are linear and in $\mathbb{R}[x]$ they aren't more than 2 degree. However,in $\mathbb{Q}[x]$ we can have an irreducible polynomial ...
2
votes
0
answers
69
views
Is the discrete logarithm equivalent to solving polynomial discrete logarithms?
Suppose we can quickly solve the discrete logarithm modulo $p$. Let's say $2$ is a generator so we can quickly find $l$ for which $2^l =h$ for any given target $h$.
An interesting observation is that ...
12
votes
1
answer
677
views
How many integrals can give multiples of $\pi$?
This question notes a few families of rational functions whose integrals (from $0$ to $1$) give rational multiples of $\pi$. A fairly straightforward explanation is given there and in the related Math....
0
votes
1
answer
94
views
Probabilistic bounds of random polynomials
This is follow-up question to my previous question about the expected number of roots .
I am considering a random polynomial given by $$p(z) = \sum_{i=0}^{n} a_i z^i$$,
where each coefficient } $a_i$ ...
0
votes
0
answers
70
views
Minimal polynomial of $\operatorname{cd}\frac{4K}{n}$
Define
$$K=\int_0^1 \frac{dx}{\sqrt{1-x^4}}$$
and the Jacobi elliptic function $\operatorname{cd}$ with modulus $i$ by
$$\int_{\operatorname{cd}z}^1\dfrac{dx}{\sqrt{1-x^4}}=z$$
on $z\in [0,2K]$ and by
...
2
votes
1
answer
114
views
Expected fraction of roots in the unit disc of random polynomial with Gaussian coefficients
I am trying to find the expected fraction of roots located in the unit disc for a random polynomial with Gaussian coefficients. Given a random polynomial
$$P(z) = a_0 + a_1 z + a_2 z^2 + \dots + a_n z^...
4
votes
1
answer
203
views
Any conjectures about Jack Littlewood-Richardson coefficients when Schur LR > 1?
Stanley famously conjectured ("Some combinatorial properties of Jack symmetric functions" Adv. in Math. (77) 1989, doi:10.1016/0001-8708(89)90015-7, MR1014073, Zbl 0743.05072) that the Jack ...