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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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
...
- 118k
0
votes
0
answers
71
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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 ...
- 24.2k
2
votes
0
answers
132
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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 ...
- 25.5k
2
votes
0
answers
150
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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 ...
- 1,762
2
votes
1
answer
261
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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
$$s_{n+1}=\frac{\sum_{j=1}^ks_{n+1-j}^2+q \sum_{j=1}^{k-1}s_{n+1-j}s_{...
- 85.1k
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votes
0
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492
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An algebraic version of the implicit function theorem for integers
$ \def \x {\boldsymbol x}
\def \a {\boldsymbol a}
\def \Z {\mathbb Z} $
The famous version of the implicit function theorem (IFT) starts with the assumption of continuous differentiability on the ...
- 295
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120
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Nilpotent elements of $(\mathbb{Z}/n \mathbb{Z})[x_1,...,x_m]/\langle f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)\rangle$
This is generalization of the univariate case
and also related to open problem.
Let $n,k,m,B>1$ be positive integers and $f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)$ be polynomials with ...
- 24.2k
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0
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Nilpotent elements of $K=(\mathbb{Z}/n \mathbb{Z})[x]/f(x)$
This is related to an open problem.
Let $n$ be integer and $f(x)$ polynomial with integer coefficients and set $K=(\mathbb{Z}/n \mathbb{Z})[x]/f(x)$.
Let $S$ be the set of degree 2 nilpotent elements ...
- 60.4k
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Is $\sum\limits_{k=1}^nk^m=S_3(n)\cdot\dfrac{P_{m-3}(n)}{N_m}$ for odd $m>1,\sum\limits_{k=1}^nk^m=S_2(n)\cdot\dfrac{P_{m-2}'(n)}{N_m}$ for even $m$?
I asked this question here
When I was in high school, I was fascinated by
$$
\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\;\...
- 207
1
vote
0
answers
274
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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 ...
- 781
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1
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133
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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 ...
- 636
4
votes
1
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116
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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 ...
- 178k
3
votes
2
answers
458
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Is a finite order automorphism of k[x,y] necessarily linear?
Let $k[x,y]$ be the polynomial ring in two variables over a field $k$ of characteristic zero. Every $k$-algebra automorphism of $k[x,y]$ is tame (e.g. the paper of McKay and Wang). It was pointed out ...
- 60.4k
7
votes
1
answer
307
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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 \...
- 23.1k
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 ...
- 229
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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,
$$u_3A_3X^3B_3 + u_2A_2X^2B_2 + ...
1
vote
0
answers
149
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Geometric construction of real root of quintic using marked ruler and compass
My question is motivated by a geometry problem about special folded rectangle:
'A rectangle with sides a, b (a<b) is folded along the line that passes through the center of the rectangle in order ...
1
vote
1
answer
236
views
Combinatorics and geometry underlying a refined Pascal matrix/Newton identities
The partition polynomials of OEIS A263633 give the coefficients of the power series/o.g.f of the multiplicative inverse (reciprocal) of a power series/o.g.f. and so give the Newton identities for ...
CommunityBot
- 1
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votes
2
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370
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What are the properties of umbra with moments $\{1,1/2,1/3,1/4,1/5,...\}$?
If we apply operator $D\Delta^{-1}$ to a function, we will get the (Bernoulli) umbral analog of the function. Particularly, applying it to $x^n$ we will get the Bernoulli polynomials $B_n(x)$. ...
- 9,312
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votes
0
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57
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The discrete orthogonal polynomials
I want a document or something that explains the following proposition:
The discrete orthogonal polynomials are the polynomial solutions of the given diference equation:
$$
\sigma(x)\Delta\nabla P_n(...
- 21
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 ...
- 1,141
4
votes
2
answers
630
views
Difference between Chebyshev first and second degree iterative methods
Consider linear equation $Au = f$.
We want to solve it with iterative method (assuming $A$ is good).
First order iterative method is:
$$
u^{k+1} = u^k - \alpha_{k+1}(Au^k - f),
$$
The second degree ...
- 688
2
votes
1
answer
189
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 ...
- 1,141
12
votes
3
answers
803
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Signed factors of harmonic polynomials
Let ${\rm Harm}_n^d$ be the space of real harmonic polynomials in $n$ variables, homogeneous of degree $d$. If $P\in{\rm Harm}_n^d$, then
$$\left(\frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\...
- 181
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votes
1
answer
750
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Find a special integer coefficients polynomial which takes small absolute value on [0,4]
The question is easy to state: Is there a non-constant $f\in\mathbb{Z}[x]$ such that for all $x\in [0,4]$, we have $|f(x)|\leq 1$? I do not know where to find a useful reference for it.
I did a few ...
- 1
2
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0
answers
61
views
Iterated chaos expansion
Using the notations from Normal Approximations with Malliavin Calculus, Chapter 2
random variables $F$ in the probability space generated by an iso-normal Gaussian family $X(h)$,
$$E[X(h)X(g)] = \...
- 613
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votes
2
answers
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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 ...
- 4,123
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votes
2
answers
759
views
What is a function field analog of Giuga's conjecture?
Giuga's conjecture (1950), which is still open and has strong numerical support, reads :
Let $n$ be a positive integer. If $1+\sum_{k=1}^{n-1}k^{n-1} \equiv 0\pmod{n}$ then $n$ is prime.
What would ...
- 1,024
2
votes
3
answers
1k
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Fastest Implementation of polynomial interpolation?
Suppose I am working over a field $\mathbb{F}$ and have $n$ points in the point-value representation $(x_0,x_1,\cdots,x_{n-1})$. What is the fastest way to do polynomial interpolation and convert this ...
0
votes
1
answer
111
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 ...
- 88.9k
1
vote
0
answers
56
views
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-...
- 2,758
17
votes
1
answer
661
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 ...
- 7,463
6
votes
2
answers
609
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{...
- 3,574
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votes
0
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35
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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
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votes
0
answers
128
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}
...
- 133
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votes
0
answers
1k
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The group of automorphisms of a polynomial ring in two variables over an integral domain
It is well-known that the group of automorphisms of a polynomial ring
$k[x,y]$, $k$ is any field, is a free product of $A(2)$ and $J(2)$
amalgamated along their intersection,
where $A(2)$ is its ...
2
votes
2
answers
154
views
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 ...
- 2,979
19
votes
3
answers
3k
views
What is the story behind the Chebyshev polynomials?
Is there anything reliable known about who actually discovered the Chebyshev polynomials and what the motivation and circumstances were?
The reason why I am interested in knowing, is that I needed a ...
- 5,632
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 ...
- 838
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votes
1
answer
240
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 $p$...
- 2,643
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*\...
- 30.7k
7
votes
1
answer
924
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 ...
- 317
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votes
1
answer
612
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 ...
- 6,730
1
vote
0
answers
109
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 ...
- 8,046
17
votes
2
answers
3k
views
Maclaurin's inequality on elementary symmetric polynomials of arbitrary real numbers
Is there a universal constant $C$ such that the following statement holds? For concreteness, you may assume $C=10000$.
Let $a = (a_1, \ldots, a_n)$ be $n$ arbitrary real numbers. For an integer $k$, ...
- 109k
-1
votes
1
answer
289
views
Must 'special' $u,v \in \mathbb{C}[x,y]$ be symmetric polynomials?
The idea for the following question came from Joachim König's last comment appearing
here, namely, the example with $u=x+y^3,v=x^3+y$.
Let $u,v \in \mathbb{C}[x,y]-\mathbb{C}$. Denote by $\alpha$ the ...
CommunityBot
- 1
4
votes
1
answer
328
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 > ...
- 61.5k
7
votes
1
answer
1k
views
Why are all these families of polynomials finally log-concave?
This started when I was examining certain families of unimodal polynomials, i.e. $\sum_{k=0}^n a_kx^k$ where $a_0\le a_1\le\cdots \le a_k\ge\cdots \ge a_n$.
(Notation: in the following, the $a_k$ ...
- 11.9k
12
votes
1
answer
680
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....
- 53.6k
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 ...
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