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.
2,679 questions
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Is there a (simple) criterion for membership to the base field of an inseparable extension?
Let $F$ be a field, let $f \in F[x]$, let $E$ be the splitting field of $f$, and let $e \in E$ be written in terms of the roots of $f$.
I'm looking for a simple way to establish if $e \in F$.
If $E/F$ ...
- 157
1
vote
0
answers
104
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Generalized identity with Stirling numbers of the second kind and falling factorials
It is known that Striling numbers of the second kind satisfy the relation
$$
\sum\limits_{k=0}^{n}{n \brace k}(x)_k = x^n.
$$
where $(x)_n$ is the falling factorials such that
$$
(x)_n = x(x-1)(x-2)\...
- 4,928
5
votes
1
answer
318
views
Non-negative coefficients polynomials
Let $n \in \mathbb N$ and $P,Q \in \mathbb R_+[x]$.
Is it true that $(x+1)^n\neq (x-2)^2 \times P(x)+(x-4)^2 \times Q(x)$ ?
I have asked, this question here (*), two weeks ago, but no answers.
(*) ...
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13
votes
1
answer
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+50
Polynomials for natural numbers and irreducible polynomials for prime numbers?
Let $p$ be a prime and $n$ be a natural number.
Define inductively for prime numbers: $f_1(x) := 1$, $f_2(x):=x$, $f_p(x) := 1+\prod_{q\mid p-1} f_q(x)^{v_q(p-1)}$.
Is $f_p(x)$ always irreducible for ...
- 3,022
0
votes
0
answers
39
views
Optimizing over convex polynomials
I have a minimization problem which reads $\min\limits_P J(P)$, where the minimum is over convex polynomials in $n$ variables, with degree at most $d$, and $J$ is a function taking polynomials as ...
- 101
1
vote
0
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70
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Quadratic forms with the same roots over GF(2) for low rank problems
Let $Q_1(x)=x^TA_1x$ and $Q_2(x)=x^TA_2x$ with $x\in GF(2)^n$, $A_i\in GF(2)^{n\times n}, i \in \{1, 2\}$. If $rank(A_1)=rank(A_2)=2$, is it possible that $Q_1(x)$ and $Q_2(x)$ can have the same roots ...
- 63
5
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1
answer
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Number of roots of a quadratic form over GF(2)
If $Q(x) = x^T A x$ with $x \in GF(2)^n$ and $A \in GF(2)^{n \times n}$, is there a way to find how many roots $Q(x)$ has based on any properties of $A$ (e.g., rank, etc.)?
- 63
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0
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63
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Relation between $-2/d$-norm and polynomial discriminant
Consider a homogeneous bivariate polynomial $f(x, y) = a_d x^d + a_{d-1} x^{d-1} y + \cdots + a_0 y^d$ of degree $d > 2$, and consider the “$-2/d$-norm”
$$\int_0^{2\pi} |f(\cos{\theta}, \sin{\theta}...
- 111
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0
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117
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Uncomplete argument in Nishioka book
In Nishioka book "Mahler functions and transcendence" in the proof of Theorem 4.2.1, Nishioka asserts the following: For a matrix $A=(a_{i,j})_{1\le i\le m}$ with coefficients in $K[z]$ ($K$ ...
- 3,996
1
vote
0
answers
45
views
Proper Pisot n-tuples
Recall that x is a Pisot number if it is real and x>1, while all of its conjugates have magnitude less than 1. Then $\{(x)^k\}$ (where $\{\cdot\}$ is the fractional part of x) approaches 0 ...
- 680
0
votes
1
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Partial sums of binomial coefficients and related family of polynomials
Let $a(n)$ be A302117. Here
$$
a(n) = 4(n-1)a(n-1) - \frac{1}{3}\prod\limits_{k=0}^{n-1}(2k-3), \\
a(0) = 0.
$$
Let
$$
T(n,k) = \sum\limits_{i=0}^{k} \binom{n}{i}.
$$
Let $P_n(z)$ be the family of ...
- 4,928
8
votes
0
answers
280
views
Meaning of the Ehrhart polynomial at $-1/2$?
I am studying a large collection of lattice polytopes, all of them being simple and empty. The dimension can be any integer. The dilatation by $2$ gives non-empty polytopes.
For many of these ...
- 3,587
3
votes
2
answers
358
views
Largest prime factors of integer polynomials
I have a question in analytic number theory which is closely related to the open problem (Bunyakovsky conjecture and more generally, Schinzel's hypothesis H) that asks you if, any irreducible ...
- 302
3
votes
1
answer
191
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Density of Pisot polynomials
Recall that a Pisot polynomial $P=x^n+ a_{n-1}x^{n-1}\ldots a_1$ has integer coefficients, a real root $x_1>1$ and all other roots $|x_i|<1$ for $1\leq i \leq n$. One key result is that $\{(...
- 680
0
votes
1
answer
72
views
Relating the order of a polynomial to the resultant in the context of formal power series
I urgently need to understand how to begin or the complete proof of the following statement:$\DeclareMathOperator{\Res}{Res}$
While reading the paper here on page one, in the introduction, the author ...
1
vote
0
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50
views
Insights on non-commutative operator families on rational functions satisfying the braid relation
I am studying the article "Symmetrization operators in polynomial rings" by A. Lascoux and M.-P. Schützenberger (MSN). Specifically, I am trying to prove the following claim involving ...
- 141
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0
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271
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What is known about vector subspaces of polynomial rings closed under factors?
Let $R$ be a commutative ring. Call a nonempty subset $F$ of $R$ a factroid if it is closed under sums and factors. That is:
If $a,b \in F$, then $a+b \in F$, and
If $a,b \in R$ with $a\in R$ ...
- 1,802
4
votes
0
answers
167
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How to prove the following equation (which involves binomials and determinant of 2×2 matrices)?
I have tried many ways to prove the following equation, such as the method of induction and expanding all the terms in the summation,but things got more complicated.I could not find an appropriate ...
- 41
0
votes
1
answer
48
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How to efficiently replace the Paterson-Stockmeyer algorithm and reduce non-scalar multiplications
Paterson-Stockmeyer algorithm
If we need to compute a high-degree polynomial expression, such as:
$$
P(y) = \sum_{k=0}^{B} a_k y^k
$$
the Paterson-Stockmeyer algorithm can process the powers in ...
- 3
1
vote
1
answer
216
views
What is the fastest known algorithm for evaluating a homogeneous binary polynomial?
This question was initially posted on math.stackexchange.com, but there is no appropriate answer, hence I have the right to publish it here again.
Let $f(x,y) = \sum_{i = 0}^d f_i x^i y^{d-i}$ be a ...
- 1,833
5
votes
1
answer
377
views
Factorisation of division polynomial
Let $\Psi_n$ denote the $n$-th division polynomial associated with the elliptic curve $y^2 = x^3 + A$, where $n$ is a natural number. The division polynomials are defined recursively, as described in ...
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2
votes
1
answer
184
views
Uniqueness of differences of roots of polynomials over finite field
Let $f$ be a polynomial over a finite field $\mathbf{F}_p$ with $p \neq 2$. Let $R$ be the roots of $f$ in some extension field. I am interested in the multiset of differences $R - R = \{ r - s \mid r,...
- 299
4
votes
0
answers
179
views
Subgroups that conjugate-cover the ambient group
Let $G$ be a finite group, and suppose that a set of proper subgroups $H_1,\dotsc,H_n$ satisfy $G=\bigcup_{g\in G}\bigcup_{i=1}^nH_i^g$, where $H_i^g$ is the conjugate of $H_i$ by $g$. In this case, ...
0
votes
0
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44
views
Sufficient conditions for a homogeneous polynomial to have a continuous right inverse
this is a question that continues a series of questions I'm coming up with on homogeneous polynomials, like for example this one.
For now I can prove that a homogeneous polynomial $f:\mathbb R^n\to \...
- 311
20
votes
1
answer
616
views
Conjecture on the number of roots of $z^n + P(z)$ within the unit disk
Some other people and I have noticed that the following seems to be true.
Fix an integer polynomial $P \in \mathbb{Z}[x]$. Let $a_n$ be the number of roots of $z^n + P(z) = 0$ that lie in the unit ...
- 303
0
votes
1
answer
98
views
Only special permutations result in a constant expression when permuting coefficients in a sum involving binomials?
Fix $n\geq 1$ and let $p_k(x) := x^k(x-1)^{n-k}$.
Suppose $\pi$ is a permutation on $\{0,1,\dotsc,n\}$, such that
$$
\sum_{k=0}^n (-1)^k \binom{n}{k} p_{\pi(k)}(x) \text{ is a constant}.
$$
Must it be ...
- 15.8k
1
vote
0
answers
95
views
Two new formulas to solve the Bring quintic using only Jacobi $\vartheta_3(q)$ and $\vartheta_4(q)$?
(Note: Emil Jann Fiedler found the formula for the Bring quintic using $R(q)$ in 2021, and these two formulas using $\vartheta_3(q)$ and $\vartheta_4(q)$ in 2022.) Recall the Jacobi theta functions,
$$...
- 12.6k
1
vote
0
answers
58
views
Universal formulas for polynomials with prescribed jets
Let $A$ be a commutative ring and $f\in A[x]$ a split monic. When $f$ is separable with roots $\mathrm Z(f)= \{ a_1,\dots ,a_k \}$, the Chinese remainder theorem (CRT) ensures that evaluation is an $A$...
- 10.5k
2
votes
0
answers
89
views
The linear independence and linear elimination of non-crossing matching polynomials
Consider the polynomial set:
$$
f_{ij} = (t_i - t_j)x_i x_j + x_i - x_j, \quad (1 \leq j < i \leq 2n)
$$
where $ t_1, t_2, \dots, t_{2n} $ are pairwise distinct.
Let's look at the non-crossing ...
- 21
0
votes
0
answers
69
views
Evaluating the coprimality in a bivariate polynomial equation
Given a prime $p>2$, let $x$ and $y$ be real numbers such that $x>y>0$ and
$$ \begin{equation} x^p-y^p=(x-y)^p+pxy(x-y)R \tag{1} \label{eq:one} \end{equation} $$
where $R$ is a bivariate ...
- 125
6
votes
1
answer
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Integral points of polynomials - a Furstenburg-type "topology" on $\mathbb{Z}$
Given $S \subseteq \mathbb{C}$, define $\displaystyle \mathfrak{c}(S) = \bigcap_{p(x) \in \mathbb{C}[x] \wedge p(S) \subseteq \mathbb{Z}}p^{-1}(\mathbb{Z}) \supseteq S$ ("the integral points ...
- 1,543
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votes
0
answers
54
views
What properties do these "norm-equal" polynomials have?
Let us first define the "norm-equal" polynomials :
For $f(x),g(x)\in \mathbb{C}[x]$, if $\forall z\in \mathbb{C},|z|=1$, we have $|f(z)|=|g(z)|$, then we call $f(x)$ and $g(x)$ are "...
- 19
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0
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A possible application of deformation theory?
Let $f : \mathbb{R}^{n} \to \mathbb{R}$ be a real-valued polynomial function. Consider the family of real algebraic sets:
$$
V_c = f^{-1}(c), \quad c \in (-1,1).
$$
I am interested in determining how ...
- 357
2
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1
answer
198
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Maximal sub-$\mathbb{C}$-algebras of $\mathbb{C}[x,y]$
After asking this question and finding this relevant paper, I would like to ask the following question:
For every $a,b \in \mathbb{C}$, denote:
$A_{a,b}=\mathbb{C}[(x-a)(x-b),x(x-a)(x-b),y]$
and
$B_{a,...
- 2,837
0
votes
0
answers
31
views
Zero sets of sums of multivariate polynomials defined recursively with mutually disjoint support
It is difficult in general to say anything about the zero sets of a sum of two or more multivariable polyomials. However, I am interested in two special cases:
I have a family of multivariate ...
- 357
3
votes
1
answer
239
views
A palindromic formula for simple convex polytopes
Let $P$ be a simple convex $d$-polytope (a $d$-dimensional convex polytope in which the number of edges incident to a vertex is $d$) and let $n_i$ be the number of $i$-faces of $P$. Is it true that ...
- 161
2
votes
0
answers
113
views
Numbers of positive terms in polynomials equal A069999
Let $a(n)$ be A069999 (i.e., number of possible dimensions for commutators of $n \times n$ matrices; it is independent of the field). OEIS states that no generating function is known.
Let $P(n,k)$ be ...
- 4,928
7
votes
2
answers
186
views
Non-locally connected polynomial Julia sets
What are some examples of complex polynomials whose Julia sets are connected, but not locally?
In the book Complex Dynamics by Carleson and Gamelin, I found:
They seem to reference:
But what is a ...
- 3,317
0
votes
0
answers
61
views
Bounding the coefficients of a polynomial written as the sum of powers of linear forms
Crosspost: Just a heads up, I've posted this question on math.stackexchange as well. I have made new attempts at solving it though, and figured I'd ask here since it's more of a research-level ...
- 121
2
votes
1
answer
498
views
When is product of polynomial and power series a polynomial?
I am currently pondering over the following question:
For a field $K$, denote by $K[X]$ and $K[[X]]$ the ring of multivariate polynomials and the ring of multivariate power series, respectively.
Given ...
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4
votes
0
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267
views
If $\mathbb{C}[a,b,c] \subsetneq \mathbb{C}[x]$, then there exist $f,g$ s.t. $\mathbb{C}[a,b,c] \subseteq \mathbb{C}[f,g] \subsetneq \mathbb{C}[x]$
I ran into this MSE question and would like to ask about its answer and plausible generalizations.
The quoted MSE question asks if the following claim is true or false and why:
Claim: Let $a,b,c \in \...
- 2,837
0
votes
0
answers
60
views
Algorithm for $q$-Bell numbers
Let $T(n,k)$ be A126347 (i.e., triangle, read by rows, with row polynomials $B(n, q)$). Here
$$
B(n, q) = \sum\limits_{k=0}^{n-1}\binom{n-1}{k}B(k, q)q^k, \\
B(0, q) = 1.
$$
Start with vector $\nu$ of ...
- 4,928
0
votes
0
answers
55
views
Find the solution of the eighth-degree polynomial equation with symmetrical structures
To solve the given equation:
$$P_1 + P_3 + P_4 + P_5 = P_2$$
where
\begin{align}
P_1 & = \prod_{i=1,2,3,4;j=1,2}(s-s_{ij}), \\
P_2 & = a\cdot \prod_{i=1,2;j=1,2}(s-s_{ij}), \\
P_3 & = b \...
7
votes
0
answers
208
views
How biased is $(x_i x_j)_{i,j}$, $x_i\in \mathbb{F}_2$?
Let $N = \frac{n (n-1)}{2}$. Let $V$ be the $N$-dimensional vector space over $\mathbb{F}_2$ consisting of tuples $(x_{(i,j)})_{1\leq i <j \leq n}$, $x_{(i,j)}\in \mathbb{F}_2$. Let $S$ be the set ...
- 20.1k
2
votes
0
answers
82
views
Beyond the Bring Radical: What is known about "generating radicals" for roots of polynomials of a given degree?
Famously, there is no general solution by radicals to find roots of polynomials (real, say) with degree $d\geq 5$. Somewhat less famously, there is a general solution[?] in degree $5$ using the so-...
0
votes
2
answers
282
views
Can a variety be the graph of a function in more than one way?
Let $V\subset \Bbb R^n$ be an irreducible affine variety of degree $\ge 2$ and $U_V\subseteq V$ a (Euclidean) open subset. Suppose that $U_V$ is the graph of a rational function, that is, there is an ...
- 13.6k
1
vote
0
answers
70
views
When a surjective homogeneous polynomial map is not open at the origin
Question(s):
Do there exist surjective homogeneous polynomial maps $f:\mathbb R^2\to \mathbb R^2$ (A) of odd degree less than $5$, and (B) of even degree less than $8$, such that the origin is not ...
- 60.5k
6
votes
1
answer
315
views
Sum of derivative of polynomial over its simple roots
Let $P$ and $Q$ be polynomials over $\mathbb C$, and $n\in\mathbb N$ be a positive integer. I'm interested in the root sums of the form
$$ \sum_{P(x)=0}\frac{Q(x)}{P'(x)^n},$$
where the sum runs over ...
- 99
2
votes
1
answer
314
views
Are surjective homogeneous maps open at zero?
I'm asking this question as a follow-up inspired by this one: An open mapping theorem for homogeneous functions?
I'm actually wondering whether there exists an homogeneous map $f:\mathbb R^n\to\mathbb ...
- 311
3
votes
0
answers
212
views
A family of polynomials related to integer partitions
For a positive integer $n$, let $p(n)$ be the number of partitions of $n$.
For $1\le k\le n$, let $p(n,k)$ denote the number of partitions of $n$ having exactly $k$ terms; in other words, $p(n,k)$ is ...
- 15.6k