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.
2,826 questions
3
votes
1
answer
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Sequences of irreducible polynomials
During some digging of mine, I once found the following recursively defined family of polynomials: $P_0=P^2+2; P_{k+1}=P_k^2-2$. Using them one can show with purely algebraic means that the 2-adic ...
18
votes
2
answers
915
views
Irreducibility of a family of integer polynomials
"Let $f(x)$ be a polynomial of degree at least 2 with $f(\mathbb{N})\subset \mathbb{N}$. Then the set of natural numbers $n$ such that $f(x)-n$ is reducible over $\mathbb{Q}$ has density 0."
...
- 345
6
votes
1
answer
230
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Global to local solutions of $x^n-y=0$
Let $n,y$ be positive integers that are greater than 1. Suppose that $x^n=y$ does not have a solution in positive integers. Then is it true that for infinitely many primes $p$ there are no solutions $...
- 345
21
votes
1
answer
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Polynomial taking only 0 and 1 values at many consecutive integers
The interpolation already gives a polynomial $f(x)$ of degree at most $n$ that makes $(f(0),f(1),\dotsc, f(n))$ attain any real sequence in $\mathbb{R}^{n+1}$.
I'm curious about the following:
Is ...
- 353
5
votes
0
answers
144
views
Solving a large system of polynomial equations over $\mathbb{F}_2$
In my work I commonly encounter large systems of polynomial equations for which it would be useful to know if there is a nontrivial solution over $\mathbb{F}_2$ (and if so, to find such a solution).
...
- 19.4k
17
votes
0
answers
360
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Minimal number of variables needed to parametrize $\operatorname{SL}_2(\mathbb{Z})$
Recently I am studying the paper
Leonid Vaserstein, Polynomial parametrization for the solutions of Diophantine equations and arithmetic groups, Annals of Mathematics 171 issue 2 (2010) pp. 979–1009, ...
- 30.7k
6
votes
0
answers
141
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Is this Hankel matrix involving Bernoulli polynomials positive definite?
Let $(B_n(X))_{n \ge 0}$ denote the sequence of Bernoulli polynomials. For any couple of integers $(d,m) \in \mathbb{N}$, define the Hankel matrix
$$ H_d(m) := \left( \frac{B_{i+j+1}(m)}{i+j+1} \right)...
1
vote
0
answers
141
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Lexicographically maximal vanishing sums of $n$-th roots of unity
Let $n$ be a positive integer with $6\mid n$, and let
$$
\zeta := e^{2\pi i / n}.
$$
For a given integer $m\in\{2,\dots,n-2\}$, consider subsets
$$
A \subset \{0,1,\dots,n-1\}
$$
of size $|A|=m$ ...
- 55
6
votes
1
answer
367
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Divisibility relations among degrees of irreducible factors of a binomial
Suppose $K$ is an algebraic number field, and $a \in K$. Let $n$ be a positive integer. The polynomial $t^n - a \in K[t]$ splits as a product of irreducible factors of degrees $d_1, \dots, d_r$.
Is it ...
- 181
4
votes
0
answers
131
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Characterize nonzero integers via a polynomial in two variables
In a paper published in 1985, Shih-Ping Tung observed that an integer $m$ is nonzero if and only if $m=(2x+1)(3y+1)$ for some $x,y\in\mathbb Z$. In fact, we can write a nonzero integer $m$ as
$\pm3^a(...
- 18.1k
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votes
0
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50
views
Linear-Disjointness of the field obtained upon iterated pre-images
Suppose $K$ is a number field and we have finitely many elements $\alpha_{1}, \ldots, \alpha_{k} \in K$ and a polynomial $f(x)\in K[x]$ of degree $\geq 2$. Given $m \geq 1$, we define $K_{m}(\alpha_{i}...
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1
vote
0
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Zeros of the partial sums $\sum_{k=0}^n (-1)^k/(z-k)$
let
$$
D_n(z)=\sum_{k=0}^n \frac{(-1)^k}{z-k}
=\frac{P_n(z)}{Q_n(z)}, \qquad
Q_n(z) = \prod_{k=0}^n (z-k).
$$
We have $\deg P_n = n$ ($n$ even), and $\deg P_n = n-1$ ($n$ odd). Moreover, using the ...
- 703
2
votes
1
answer
317
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How close can general algebraic curves get to rational curves?
Let $K$ be a field (one may assume that $K$ has characteristic zero and is algebraically closed, if this helps), and let $f \in K[x,y]$. Suppose that the curve $C_f : f(x,y) = 0$ is not a rational ...
- 30.7k
3
votes
0
answers
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Many degree $d$ nilpotent elements of quotients of polynomial rings and non-vanishing product
Generalization of this question.
Let $n$ be positive integer and $f_1(x_1,...,x_k), \dots, f_m(x_1,...x_k)$
polynomials with integer coefficients.
Let $K=\mathbb{Z}/n\mathbb{Z}[x_1,...,x_k]/\langle ...
- 25.8k
2
votes
1
answer
681
views
A question about positive polynomials
Are there some $P,Q \in \mathbb R_+[x]$ with $(x+10)^{2025}=(x+2025)^2P(x)+(x+2024)^2Q(x)$ ?
PS : the AI give an negative answer in the case $(x+1)^{2025}$
I have posted the question here (*), but no ...
- 6,010
2
votes
0
answers
94
views
Can $w^2+bx^2+cy^2+dz^2$ be universal over a sparse subset of $\mathbb N$?
Let $b,c,d\in\mathbb N=\{0,1,2,\ldots\}$ with $1\le b\le c\le d$. For a subset $S$ of $\mathbb N=\{0,1,2,\ldots\}$, if
$$\{w^2+bx^2+cy^2+dz^2:\ w,x,y,z\in S\}=\mathbb N$$
then we say that $w^2+bx^2+cy^...
- 18.1k
2
votes
0
answers
85
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About shapes of bivariate polynomials
Let $f \in \mathbb{C}[x,y]$. It is known that $f$ can be factored into Puiseux series. Indeed, if we write
$$\displaystyle f(x,y) = \sum_{j=0}^n a_j(x) y^j,$$
then we can obtain a factorization of the ...
- 30.7k
1
vote
0
answers
53
views
Effective bounds for degree and height in algebraic number enumeration
I am considering a specific enumeration of all complex algebraic numbers within the unit ball, defined as follows:
Enumerate all primitive, irreducible polynomials in $\mathbb{Z}[x]$ with positive ...
- 545
2
votes
0
answers
89
views
Asymptotic of dimensions of subvarieties of linear spaces that are nearly norm-dense in the unit balls
This is a follow-up to this question. I will start with the problem statement first (here, $B_n$ denotes the closed Euclidean unit ball of $\mathbb{R}^n$):
Fix sufficiently small $\varepsilon > 0$....
- 5,085
2
votes
0
answers
183
views
Irreducibility of a degree-$27$ polynomial through Newton polygon or residual reduction
Let $K=\mathbb{Q}_3(t)$ be the finite extension of the $3$-adic number field $\mathbb{Q}_3$, where $t=3^{1/13}$. I want to know if the polynomial $$f(x)=(x^9-t^2)^3-3^4x+3^3t^5 \in K[x]$$ is ...
- 450
3
votes
1
answer
392
views
Question about common zeros of hypersurfaces in projective space
Let $V$ be a vector space of homogeneous polynomials of degree $d$ in $n$ variables, or equivalently a subspace of the linear system of degree $d$ hypersurfaces in projective space $\mathbb{P}^{n-1}(\...
- 23.7k
1
vote
0
answers
52
views
Degree of multidimensional resultant arising from a "traceless" polynomial decomposition
In the course of my PhD project, I have encountered the following problem concerning multidimensional resultants.
I am interested in characteristic polynomials of traceless matrices, i.e., univariate ...
- 11
0
votes
0
answers
37
views
$O(1)$ algorithm for factoring integers of the form $n=X (X^D+O(X^{D-1}))$
Factorization of integers of special forms are of both theoretical
interest and cryptographic implications.
Experimentally we found a seemingly "large" set of integers for which a divisor ...
- 25.8k
1
vote
0
answers
107
views
Determinantal elimination for $f_i=x_i(y+t_i)-1$: is there an analogue for $f_i=x_i(y+t_i z+s_i)-1$?
Consider the polynomials
$$
f_i = x_i (y + t_i) - 1,
$$
where the variables are $x_i$ and $y$.
Experiments indicate that, if the coefficients $t_i$ are algebraically independent, then there exists a ...
2
votes
0
answers
110
views
Calculating Polynomial Resultants Quickly
I'm working on a project that requires quickly calculating the resultant of two polynomials in $\mathbb{Z}[x]$ of large degree $d,e$. On the Wikipedia article for polynomial resultants, it says that
...
- 181
4
votes
1
answer
262
views
Is $\mathbb{Z}_{>0}$ diophantine in $(\mathbb{Q}_{>0},+,\cdot,1)$?
It is well known that we can define $\mathbb{N}$ in $(\mathbb{Z},+,\cdot)$ via an existentially quantified equality, as follows. Letting $n$ be an integer parameter
$$
n\in \mathbb{N} \...
- 19.4k
0
votes
0
answers
178
views
On quadratic equations over the Gaussian ring $\mathbb Z[i]$
In 1972 C. L. Siegel proved that there is an algorithm to decide for any polynomial $P(x_1,\ldots,x_n)\in\mathbb Z[x_1,\ldots,x_n]$ with $\deg P=2$ whether
$$P(x_1,\ldots,x_n)=0$$ for some $x_1,\ldots,...
- 18.1k
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votes
0
answers
62
views
Divided differences of TP kernels yield distinct, real-rooted polynomials
Let $\{w_j:~1\le j\le N\}$ be a set of non-zero real numbers with $\sum_{j} \frac{1}{|w_j|}<\infty$. We define a polynomial $P(\xi,z)=\sum_{k=0}^{N-1}f_s(\xi)z^{s}$, where $f_s(\xi)$ is a real ...
- 9
7
votes
0
answers
280
views
Polynomial identification of natural numbers
Let $n$ be an integer parameter. Is there a polynomial $f(x,y,n)\in \mathbb{Z}[x,y,n]$ such that
$$
n\in \mathbb{N} \Longleftrightarrow \exists x,y\in \mathbb{Z}, f(x,y,n)=0?
$$
This is a generalized ...
- 19.4k
1
vote
0
answers
122
views
On the largest Eigenvalue of a certain "graph Laplacian"
I am interested in the following problem, which came up while working on this paper about estimating Betti numbers of Kähler manifolds, where we were not able to solve it and had to resort to ...
- 1,276
5
votes
1
answer
210
views
Remez-type inequality
This question is about a statement on a Remez-type inequality from the paper Polynomial Inequalities on Measurable Sets and Their Applications by M. I. Ganzburg (Constr. Approx. (2001) 17: 275–306).
...
- 53
0
votes
1
answer
133
views
Must $x (x^D+1) -n=0$ have at most one root over the integers for $D>1$?
Let $D>1,n>0$ be integers. For all $D,n$, must $x (x^D+1) -n=0$ have at most one integer root?
Experimental data for $D \in \{2,3\}$ and $n=x_0 (x_0^D+1)$ there is only one solution for $1 < ...
- 25.8k
0
votes
0
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158
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On factoring integers of the form $n=(x^D+1)(y^D+1)$ and $n=(x^D+a_{D-2}x^{D-2}+\cdots+ a_0) (y^D+b_{D-2}y^{D-2}+... b_0)$ with $x,y$ of the same size
From our preprint On factoring integers of the form $n=(x^D+1)(y^D+1)$ and $n=(x^D+a_{D-2}x^{D-2}+\cdots+ a_0) (y^D+b_{D-2}y^{D-2}+ \cdots+b_0)$ with $x,y$ of the same size.
We got plausible ...
- 25.8k
1
vote
0
answers
119
views
Heuristics for spectral norm of directed adjacency matrix connected to prime numbers?
Are there any heuristics to compute the spectral norm of the adjacency matrix of this directed graph connected to prime numbers?
Let $p$ be a prime and $n$ be a natural number.
Define inductively for ...
3
votes
0
answers
112
views
Link between Schur polynomials and generators of the Virasoro algebra
I am currently studying Schur polynomials in the context of a representation of the Virasoro algebra for bosons (with central charge $c=1$). The generators of the algebra are denoted $L_k^{(n)}$, in ...
- 131
2
votes
0
answers
100
views
Regularity of a variant of elementary symmetric polynomials
Let $Y_1, Y_2,\dots, Y_n$ be $n$ bases of linear forms in the polynomial ring $k[x_1,\dots, x_n]$, where $k$ is a field or $\mathbb{Z}$. Examples of bases $Y_i$'s are $A_i\cdot[x_1,\dots, x_n]^T$, ...
- 21
7
votes
0
answers
626
views
Linear independence of composition of polynomials
Let $k, d \in \mathbb{Z}_{\geq 1}$ be positive integers.
I want to prove the following.There exists a large enough $r \in \mathbb{Z}_{\geq 1}$ (depending on $k$ and $d$) and a Zariski open set $U \...
- 409
3
votes
1
answer
168
views
Equivalence of continuous and discrete $L^{1/2}$ "norms"
Recall the iterated Trapezoidal rule of quadrature:
$$
\int_0^1 f(x) \, dx \approx I_n f := {1 \over 2n} \left(f(0) + f(1) + \sum_{k=1}^{n-1} 2f(k/n) \right).
$$
Recall also the $L^{1/2}$ "norm&...
- 1,051
5
votes
0
answers
184
views
Can we construct a polynomial $T$ such that $g(\alpha)$ is a root of $T$, for all $\alpha \in Z(f)$, where $g, f$ are two monic polynomial?
Let there be two monic polynomials $f(x), g(x) \in \mathbb{Z}/(p^k \mathbb{Z})[x]$, where $p$ is an odd prime number and $k \geq 2$. Can we construct a polynomial $T$ such that $g(\alpha)$ is a root ...
- 311
3
votes
1
answer
318
views
Converse of the product property for palindromic polynomials
Assume that $p(x)$ is a polynomial in $\mathbb{Z}_{>0}[x]$ with a factorization
$$
p(x) = p_1(x)\cdots p_m(x),
$$
where each $p_i(x)\in \mathbb{Z}_{>0}[x]$, not necessarily irreducible.
We say ...
- 1,365
4
votes
1
answer
410
views
Generalizing Marden's theorem to quartics
Is there a nice generalization of Marden's theorem which applies to all quartics?
Marden's theorem is a strengthening of the Gauss-Lucas theorem for polynomials over the complex numbers which applies ...
- 8,540
1
vote
0
answers
141
views
Solving equations on a high dimensional torus
I want to establish some useful criteria for uniqueness of solutions to the following:
$$Mx=b,\\ \text{subject to}\ ||x(2k-1:2k)||=1, k=1,2,\cdots,5,$$ where $M\in\mathbb{R}^{10\times10},\ x\in\mathbb{...
- 81
0
votes
0
answers
238
views
Coercive polynomials of small order of coercivity
I am following the definition and notation in this paper: How fast do coercive polynomials grow?
In particular, we say that a polynomial $f \in \mathbb{R}[x_1, \cdots, x_n]$ is $q$-coercive if for all ...
- 30.7k
3
votes
1
answer
336
views
On the polynomial $x^{n-1}+\frac{x^{n-2}}2+\cdots+\frac1n$
Motivated by Question 498655, here we introduce the polynomials
$$S_n(x):=\sum_{k=1}^n \frac{x^{n-k}}k=x^{n-1}+\frac{x^{n-2}}2+\cdots+\frac1n\ \ \ (n=1,2,3,\ldots),\tag{1}$$
which are related to the ...
- 18.1k
0
votes
0
answers
143
views
Integer solutions of bivariate polynomial equations
Given a polynomial $f(x,n)$ with integer coefficients, I want to find all pairs of rational number $x$ and positive integer $n$ such that $f(x,n)=0$. The polynomials I'm looking at are like the ...
- 176
0
votes
0
answers
139
views
Is this ideal of a polynomial ring over a field a contracted ideal?
Let $k$ be a field and $R:= k[y_1, \dotsc , y_d]$ be a polynomial ring in $d$ variables over $k$. Set $K:= QF(R)$. Given finitely many elements $a_1, \dotsc , a_n$ algebraic over $K$, we consider the ...
- 101
1
vote
2
answers
312
views
Expansion identity for the Eulerian polynomials of the second order
Background
$\newcommand{\polylog}{\mathrm{PolyLog}}$
The Eulerian polynomials $A_{m}(\cdot)$ are defined by the exponential generating function:
\begin{equation}
\frac{1-x}{1-x \exp[ t(1-x) ] } = \...
- 5,065
2
votes
2
answers
245
views
$\left\{\frac{x(ax+b)}2+\frac{y(ay-b)}2:\ x,y=0,1,2,\ldots\right\}$ and asymptotic bases of order 2
A subset $A$ of $\mathbb N=\{0,1,2,\ldots\}$ is called an asymptotic base of order $h$ if any sufficiently large $n\in\mathbb N$ belongs to the set
$$hA=\{a_1+\ldots+a_h:\ a_1,\ldots,a_h\in A\}.$$
...
- 18.1k
0
votes
0
answers
197
views
Practical partial fraction decomposition
I have tried to implement Ramanujan's algorithm for Solvability of a system of polynomial equations but got stuck in the final step of calculating the partial fraction decomposition from which the ...
- 14k
0
votes
1
answer
107
views
Closed forms or special function representations for certain binomial sums involving harmonic-like terms?
I’m studying the following family of polynomials defined for integers $n \geq 1$:
\begin{aligned}
A_n(x) &= \frac{x^n}{(n-1)!} \left[
\sum_{k=0}^{\left\lfloor \frac{n-1}{2} \right\rfloor} \binom{n-...
