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.
2,819 questions
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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(...
- 18k
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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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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
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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 ...
- 30k
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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 ...
- 25.7k
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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 ...
- 6,007
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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^...
- 18k
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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 ...
- 30k
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0
answers
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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
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80
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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
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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
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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 $\mathbb{P}^{n-1}(\...
- 23.7k
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Degree of multidimensional resultant arising from a "traceless" polynomial decomposition
In the course of my PhD project, I have encountered the following problem concerning multidimensional resultants.
I am interested in characteristic polynomials of traceless matrices, i.e., univariate ...
- 11
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$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 ...
- 25.7k
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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 ...
