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.
2,486
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Is $\sum\limits_{k=1}^nk^i=S_3(n)\times\frac{P_{i-3}(n)}{N_i}$ for odd $i>1,\sum\limits_{k=1}^nk^i=S_2(n)\times\frac{P_{i-2}'(n)}{N_i}$ for even $i$?
I asked this question here
When I was in high school, I was fascinated by $\displaystyle\sum\limits_{k=1}^n k= \frac{n(n+1)}{2}$ so I tried to find the general solution for $\displaystyle\sum\...
- 103
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172
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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 ...
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1
answer
96
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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 ...
- 21
2
votes
1
answer
222
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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}^{...
- 143
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votes
1
answer
109
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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 ...
- 173
7
votes
1
answer
263
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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 \...
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68
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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 ...
- 229
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68
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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,
$$u_3A_3X^3B_3 + u_2A_2X^2B_2 + ...
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127
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Geometric construction of real root of quintic using marked ruler and compass
My question is motivated by a geometry problem about special folded rectangle:
'A rectangle with sides a, b (a<b) is folded along the line that passes through the center of the rectangle in order ...
4
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240
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An algebraic version of the implicit function theorem for integers
$ \def \x {\boldsymbol x}
\def \a {\boldsymbol a}
\def \Z {\mathbb Z} $
The famous version of the implicit function theorem (IFT) starts with the assumption of continuous differentiability on the ...
- 345
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43
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The discrete orthogonal polynomials
I want a document or something that explains the following proposition:
The discrete orthogonal polynomials are the polynomial solutions of the given diference equation:
$$
\sigma(x)\Delta\nabla P_n(...
- 21
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0
answers
130
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Proof that a specific cubic equation has three real roots [migrated]
I'm trying to prove that the cubic equation
$a_3 \lambda^{3} + a_2 \lambda^{2} + a_1 \lambda + a_0 = 0$
has three real roots. The coefficients are
$a_3 = - 1 - \sigma - \tau - \chi$
$a_2 = -2 (\sigma +...
1
vote
1
answer
78
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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 ...
- 1,131
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1
answer
158
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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 ...
- 1,131
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0
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58
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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)] = \...
- 603
16
votes
1
answer
697
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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 ...
- 215
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1
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103
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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 ...
- 3,898
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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-...
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2
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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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637
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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 ...
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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{...
- 75
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0
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32
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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 ...
- 115
2
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0
answers
103
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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}
...
- 133
2
votes
2
answers
116
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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 ...
- 1,641
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354
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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?
...
- 99k
9
votes
1
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611
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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 ...
- 1,879
2
votes
2
answers
191
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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 ...
- 796
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votes
1
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229
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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$...
- 6,014
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1
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180
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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*\...
- 581
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votes
1
answer
563
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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 ...
- 159
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0
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102
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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 ...
- 7,981
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1
answer
626
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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 ...
- 235
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0
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64
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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 ...
- 581
13
votes
1
answer
659
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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....
- 1,540
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votes
1
answer
87
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 } $a_i$ ...
- 768
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votes
0
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59
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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
...
- 303
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votes
1
answer
107
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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
$$P(z) = a_0 + a_1 z + a_2 z^2 + \dots + a_n z^...
- 768
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0
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136
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Any conjectures about Jack Littlewood-Richardson coefficients when Schur LR > 1?
Stanley famously conjectured ("Some combinatorial properties of Jack symmetric functions" Adv. in Math. (77) 1989, doi:10.1016/0001-8708(89)90015-7, MR1014073, Zbl 0743.05072) that the Jack ...
- 163
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0
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105
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Decidability of whether two polynomial bijections generate a free group
I am wondering about the decidability of the following question:
Given two polynomial bijections $f, g$ from the real numbers to the real numbers (with say rational coefficient just to simplify what &...
- 1,045
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votes
2
answers
288
views
A doubt regarding the extended form of the Weierstrass factorization theorem
I want to represent $\sin(x)-\dfrac{1}{\sqrt{2}}$ as a product of it's zeroes
According to the Weierstrass factorization theorem, the sine function can be represented as a product of its factors:
$$\...
- 103
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0
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71
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A problem on the factorizability of certain a boolean multivariate polynomial
Let $N$ be a positive integer and let $[N] := \{1,2,\ldots,N\}$. Given a non-empty collection $K$ of a subsets of $[N]$ and field $\mathbb F$, define a polynomial $f_K$ over $\{-1,+1\}$ in ...
- 6,586
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0
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250
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Cosine Modulo $p$?
Consider the integers modulo a prime $p$. I'm looking for a nice polynomial function that acts as a sort of "cosine" on the integers modulo $p$. Specifically, I'm looking for solutions to ...
- 581
2
votes
1
answer
167
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Minimal degree of a polynomial such that $|p(z_1)| > |p(z_2)|, |p(z_3)|, ..., |p(z_n)|$
I was investigating the behavior of $p(x)^n \mod {q(x)}$, for some polynomials $p, q \in \mathbb{C}[x]$. We'll assume $q$ is squarefree. If $q(x) = (x - z_1) (x - z_2) (x - z_3) ... (x - z_n)$ for ...
- 1,935
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0
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83
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Linear Program Optimal Value
If $f(A,b,c)$ is the optimal value of a linear program
$\min c.x$
subject to $A.x \leq b ; x \geq 0.$
Does $f(A,b,c)$ have a piecewise polynomial/rational upper bound in $(A,b,c)$ on the domain of ...
3
votes
1
answer
326
views
GCD in $\mathbb{F}_3[T]$ with powers of linear polynomials
This is a continuation of my previous question on $\gcd$s of polynomials of type $f^n - f$.
Let us call $n > 1$ simple at a prime $p$ when $p-1 \mid n-1$ but $p^k - 1 \not\mid n-1$ for all $k > ...
2
votes
0
answers
60
views
Number of solutions of overdetermined quadratic polynomial equations
Given $m$ linearly independent quadratic polynomials over the complex field in $n$ variables with $m>n$ and such that the number of zeros, say $N$, is finite, is there a known or conjectured strict ...
- 1,145
1
vote
0
answers
42
views
Ultra-operations numbers (polynomials) [closed]
After Bring's root article, I became interested in understanding the theory of ultra numbers and their operations. There are very few vague concepts about these numbers on the Internet. I would be ...
- 11
0
votes
0
answers
53
views
Extracting implications in polynomial constraint system from Groebner basis
Given a Groebner basis for a system of polynomial constraints over $\mathbb{Q}$, are there any known methods for extracting the low degree factorable polynomials in the ideal generated by that basis?
...
- 101
6
votes
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 ...
- 6,122
5
votes
0
answers
87
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Ratio of theta functions as roots of polynomials
I already asked the same question here, but received no answer. I did some little progress and so I'm asking again.
I was playing with the theta functions with argument $ z = 0 $
$ \vartheta_2(q) =\...
- 191