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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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 ...
Tom Copeland's user avatar
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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}, ...
Oleksandr  Kulkov's user avatar
36 votes
6 answers
3k views

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 ...
Denis Ivanov's user avatar
-5 votes
1 answer
69 views

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
Luật Trần Văn's user avatar
3 votes
1 answer
130 views

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 ...
Learning math's user avatar
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 ...
joro's user avatar
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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 ...
qifeng618's user avatar
  • 838
3 votes
1 answer
75 views

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 ...
Monk's user avatar
  • 125
2 votes
0 answers
129 views

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 ...
Stanley Yao Xiao's user avatar
2 votes
0 answers
146 views

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 ...
Jakub Konieczny's user avatar
0 votes
0 answers
120 views

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 ...
joro's user avatar
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4 votes
0 answers
166 views

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 ...
joro's user avatar
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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_{...
Roland Bacher's user avatar
1 vote
0 answers
220 views

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\;\...
pie's user avatar
  • 139
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 ...
Sayan Dutta's user avatar
2 votes
1 answer
127 views

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 ...
dankane's user avatar
  • 21
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}^{...
wsz_fantasy's user avatar
4 votes
1 answer
115 views

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 ...
stillconfused's user avatar
7 votes
1 answer
295 views

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 \...
Ivan's user avatar
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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 ...
G. Fougeron's user avatar
2 votes
0 answers
78 views

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 + ...
Septimus Boshoff's user avatar
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 ...
Mikhail Gaichenkov's user avatar
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 ...
Mohsen Shahriari's user avatar
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(...
Karim's user avatar
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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 ...
Oleksandr  Kulkov's user avatar
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 ...
Oleksandr  Kulkov's user avatar
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)] = \...
Julian's user avatar
  • 613
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 ...
Yanlong Hao's user avatar
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 ...
ABB's user avatar
  • 3,898
1 vote
0 answers
55 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-...
Maulik's user avatar
  • 111
6 votes
2 answers
1k views

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 ...
user108580's user avatar
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 ...
ghc1997's user avatar
  • 763
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{...
Er Bu's user avatar
  • 75
0 votes
0 answers
34 views

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 ...
Made's user avatar
  • 115
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} ...
dmitry's user avatar
  • 133
2 votes
2 answers
148 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 ...
Paul Cusson's user avatar
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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? ...
Fedor Petrov's user avatar
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 ...
Luis Ferroni's user avatar
  • 1,879
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 ...
qifeng618's user avatar
  • 838
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 $p$...
Keivan Karai's user avatar
  • 6,064
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*\...
mtheorylord's user avatar
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 ...
Hao Yu's user avatar
  • 185
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 ...
Joonas Ilmavirta's user avatar
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 ...
jdhejw's user avatar
  • 317
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 ...
mtheorylord's user avatar
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....
Steven Stadnicki's user avatar
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 } $a_i$ ...
AgnostMystic's user avatar
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 ...
Nomas2's user avatar
  • 303
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^...
AgnostMystic's user avatar
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 ...
Ryan Mickler's user avatar

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