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.
822
questions with no upvoted or accepted answers
6
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400
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Using the Lorentz operators to build polynomials that converge to a continuous function
Questions
Let $f(\lambda):[0,1]\to (0,1)$ have a $\beta-\lfloor\beta\rfloor$)-Hölder continuous $\lfloor\beta\rfloor$-th derivative, where $\beta>0$.
Find explicit bounds, with no hidden constants,...
6
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0
answers
204
views
Parameter independence of Stanley's "content formula". Why?
For a cell $\square$ in the Young diagram of a partition $\lambda$, let $h_{\square}$ and $c_{\square}$ denote the hook length and content of $\square$, respectively.
R. Stanley remarked following ...
6
votes
0
answers
736
views
Discriminant of $\alpha P(u) + (z-u) P'(u)$
I'm trying to find a “closed form” of $\textrm{Discriminant}_u(f(u))$, where $f(u) := \alpha P(u) + (z-u) P'(u)$.
Here $P(u)$ is a monic polynomial of degree $d > 1$ with $u\in\mathbb{C}$, $\alpha$ ...
6
votes
0
answers
360
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Are all trigonometric polynomials from the 3-torus to the 3-sphere homotopically trivial?
I'm looking at maps from the 3-torus $\mathbb{T}^3\simeq (\mathbb{R}/2\pi\mathbb{Z})^3$ to the 3-sphere $\mathbb{S}_3\subset \mathbb{R}^4$.
I understand that, according to Hopf theorem, continuous ...
6
votes
0
answers
321
views
Galois groups associated to matrices
When $A\in M_n(\mathbb{Q})$, we consider the pencil $A-xA^T$. Then $p_A(x)=\det(A-xA^T)$ is a self-reciprocal polynomial. $p_A$ can only be irreducible if $n=2p$ is even.
Question: For every $p$, does ...
6
votes
0
answers
321
views
Irreducibility of a palindromic polynomial
I have strong reasons to believe that the palindromic polynomial $p_n(x)$ defined by
$$p_n(x) = x^{2n}+2x^{2n-1}+3x^{2n-2}+ \cdots+ nx^{n+1}+(n+3)x^{n}+nx^{n-1}+\cdots+2x+1$$
is irreducible in $\...
6
votes
0
answers
166
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Characteristic polynomials of Cartan matrices of Lie algebras
Let $L$ be a simple Lie algebra over $\mathbb{C}$ with Cartan matrix $C_L$ (as in https://de.wikipedia.org/wiki/Cartan-Matrix )
Question 1: Is is true that the characteristic polynomial $f$ of $C_L$ ...
6
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0
answers
99
views
Class number of certain polynomials
Let $f_n(x)=x^n-\sum\limits_{i=0}^{n-1}{x^i}$ and $A_n$ the number field corresponding to $f_n$.
Question: Is the class number of $A_n$ always equal to one, or equivalently, is the ring of integers ...
6
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0
answers
111
views
A question about the span of a sequence of polynomials satisfying a linear recurrence
Let F be a finite field and A(n) in F[t], n in N, be defined by a linear recurrence with coefficients in F[t], together with initial conditions. Is there a decision procedure for determining whether ...
6
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0
answers
214
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Extremal polynomial majorants of $\log{|f|}$: a multivariate extension of a theorem of Carneiro and Vaaler
Carneiro and Vaaler have proved, as an application of their work on Beurling-Selberg extremal majorants, that for any non-zero complex polynomial $f(z) \in \mathbb{C}[z]$, the infimum value of the ...
6
votes
0
answers
406
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Computing remainders modulo $\prod_{i\in S} (x-x_i)$ fast using FFT
Note: Originally asked on Math StackExchange here, without an answer. Figured I should try here, since this is a more research-level question.
I am trying to implement a fast polynomial multipoint ...
6
votes
0
answers
217
views
Infinitude of cyclotomic polynomials with a certain number of terms
Let $\Phi_n$ be the $n$th cyclotomic polynomial:
$${\Phi _{n}(x)=\!\!\prod _{\substack {1\leq k\leq n \\ \gcd(k,n)=1}} \!\!\big(x-e^{2i\pi {k/n}}\big).}$$
Here is a list of the first 30 cyclotomic ...
6
votes
0
answers
262
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roots of a polynomial linked to mock theta function?
The following polynomial (after harmless factors dropped) is found in the paper entitled Mock theta functions and quantum modular forms by Folsom-Ono-Rhoades (see Theorem 1.1)
$$Q_k(z)=\sum_{n=0}^{k-1}...
6
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0
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251
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Concavity of a function implicitly defined by a polynomial
Consider the following system of $n$ equations:
\begin{equation}f_j^2 = x_j^2\sum_{i=1}^n A_{ij} f_i
\tag{$\star$}
\end{equation}
where $A_{ij}\geq 0$ are known constants and where $x_j>0$ for ...
6
votes
0
answers
257
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A weaker power mean than in Smale's mean value conjecture, but a neat result
EDIT: I am a fool, something stronger is true and I should have seen it immediately from my own proof. For any critical point which is not a zero itself we have:
$$ P_{-2}(\ldots) \ = \ \sqrt{\frac{n-...
6
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0
answers
131
views
About the properities of sum of powers of items in a polynomial
Given a polynomial $f_1=a_1x+a_2x^2+\cdots+a_{p-1}x^{p-1}\in\mathbb{Z}[x]/(x^{p}-1)$, with prime $p$, we may generate the other $p-1$ polynomials:
\begin{eqnarray*}
f_2&=&a_1^2x^2+\cdots+a_{p-...
6
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0
answers
115
views
Recursions which define polynomials?
Let $k$ be a positive integer and let
$$h(n,k,q)=\frac{1-(1+q^{k})q^{2k(n-1)+1}+q^{2}}{1-q^{2n-1}}h(n-1,k,q)-\frac{(1-q^{k(2n-3)})(1-q^{2k(n-1)})q^2}{(1-q^{2n-1})(1-q^{2n-3})}h(n-2,k,q)$$
with ...
6
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0
answers
326
views
Criteria for irreducibility using the location of complex roots
I would like to see criteria for the irreducibility of a polynomial over $\mathbb{Z}$ based (mainly) on the location of the roots of the polynomial in the complex plane. An example of such a criterion ...
6
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206
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Degree of Chebyshev polynomial necessary
In general, given $0<a<1$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[1-\frac{a}2,1+\frac{a}2]$ at every $x\in[1-a,1+a]$ and $f(0)=0$. What is minimum degree that is ...
6
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0
answers
468
views
Find a polynomial not in any ideal generated by polynomials of total degree $o(n)$
Is there an explicit nontrivial (= not a constant) polynomial $p \in \mathbb{C}[x_1, \ldots, x_n]$ such that, for any ideal $I \not= \mathbb{C}[x_1, \ldots, x_n]$ generated by $f_1, f_2, \ldots, f_m$ ...
6
votes
0
answers
209
views
Find a symmetric polynomial with a projection divisible by a known polynomial
Consider the polynomial $Q$, a homogeneous quartic in seven variables:
$$ Q(R, s_1, s_2, s_3, s_4, d_1, d_2) = \\
(d_1^2-(R+s_1+s_2-s_3-s_4)(R+s_1-s_2+s_3+s_4))\\(d_1^2-(R-s_1-s_2-s_3-s_4)(R-s_1+s_2+...
6
votes
0
answers
257
views
Are the roots of chromatic polynomials plus a fixed constant dense in $\mathbb{C}$?
Alan Sokal proved that chromatic roots are dense in the whole complex plane. I.e., if $P(G;z)$ denotes the chromatic polynomial of a finite simple graph $G$ evaluated at $z \in \mathbb{C}$, then $$\...
6
votes
0
answers
77
views
Bounding volume of cell in complement of zero set
I am given an integer polynomial $f \in \mathbb{Z}[X_1, \ldots, X_n]$ of bounded degree and bounded coefficient size. The polynomial's zero set partitions $\mathbb{R}^n$ into cells. What I am looking ...
6
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106
views
Irreducibility testing and factoring
It is a result of van Hoeij and Novicin (Algorithmica, 2012) that factoring polynomials of degree $d$ over the integers can be done in $O(d^6 + d^4 \log^2 A)$ time, where $A$ is the coefficient bound. ...
6
votes
0
answers
3k
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Prime ideals in polynomial rings over integers
Im trying to find a characterization of the prime ideals in the polynomial ring $R = \mathbb Z[X,Y]$ in two variables over the integers.
Actually I need to find the maximal ideals in quotient rings $...
6
votes
0
answers
271
views
Polynomial upper approximation with respect to the Gaussian measure
Let $f = 1_{[a,+\infty)}$ be the indicator function of a half-line. Does there exist a sequence $(P_n)$ of polynomials such that $f(x) \leq P_n(x)$ for every real $x$ and
$$ \lim_{n\to \infty} \int_{\...
5
votes
0
answers
163
views
Monotonicity of ratio of symmetric polynomials
The complete homogeneous symmetric polynomials of degree $\ell$ in $n$ variables are defined by
\begin{equation*}
h_{\ell}(x_1,x_2,\ldots,x_n) = \sum_{1 \leq i_1 \leq i_2 \leq \ldots \leq i_{\ell} \...
5
votes
0
answers
105
views
Ratio of theta functions as roots of polynomials
I already asked the same question here, but received no answer. I did some little progress and so I'm asking again.
I was playing with the theta functions with argument $ z = 0 $
$ \vartheta_2(q) =\...
5
votes
0
answers
117
views
Finding an $\mathbb{F}_q$-point on one specific intersection of quadrics
Let $\mathbb{F}_q$ be a finite field of large characteristic and $a_1, a_2, \cdots, a_n \in \mathbb{F}_q$ be some pairwise different elements. I assume that $\sqrt{-1} \in \mathbb{F}_q$. Consider the ...
5
votes
0
answers
151
views
Higher Cardano formulae in terms of $\Theta$
Consider a polynomial in one variable with complex coefficient
$$f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$
we are interested in its roots. Babylonian solved for $n = 2$, and Cardano did it ...
5
votes
0
answers
152
views
Nullstellensatz with nilpotents and $I=J(V(I))$
Let $R$ be the ring $$\mathbb{R}[t_1,t_2\ldots]/(t_1^2,t_2^2,\ldots)$$
Let $p_1,\ldots p_\ell$ be polynomials in $R[x_1,\ldots,x_n]$ whose constant terms are 0.
Let $f$ be a polynomial which is zero ...
5
votes
0
answers
160
views
Reducibility of $f(x)^{2^n}+1$ and $f(x)^{2^n}+g(x)^{2^n}$
Related to generalized Fermat numbers.
Let $f(x),g(x)$ be coprime polynomials with integer coefficients.
Assume that if $f(x)$ or $g(x)$ are of the form $h(x)^k$ then $k$ is power
of two.
Q1 Is it ...
5
votes
0
answers
98
views
Which reals are Lebesgue measures of regions in $\mathbb R^n$ defined by inequalities involving polynomials with integer coefficients?
Let $a$ be a real number. What are necessary and sufficient conditions for the existence of a positive integer $n$ and a finite set of polynomials $p_1,\ldots,p_k$ with integer coefficients in $n$ ...
5
votes
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answers
366
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Large prime factors of n²+1
Iwaniec proved (and many people extended) that the number of $n \le x$ for which $n^2+1=P_2$ (product of at most two primes) is $\gg x/\log x$. I am wondering what is known/can be proved for the ...
5
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0
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211
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Belyi functions with prescribed image of a given point
$\newcommand{\bP}{\mathbb{P}}\newcommand{\bQ}{\mathbb{Q}}$Definition. A Belyi function is a non-constant rational function $f:\bP_{\bQ}^1\to \bP^1_{\bQ}$ such that the image of any of its critical ...
5
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0
answers
192
views
Is there a polynomial version of Wilson's theorem which can avoid Cramer flavored conjectures?
Wilson's theorem states that a natural number $n > 1$ is a prime number if and only if the product of all the positive integers less than $n$ is one less than a multiple of $n$.
Is there a version ...
5
votes
0
answers
213
views
On Lehmer's conjecture for a given degree
Lehmer's conjecture states that the Mahler measure of a monic integer polynomial as at least $u:=1.176280818...$ when it is greater than 1, see for example https://en.wikipedia.org/wiki/Lehmer%...
5
votes
1
answer
528
views
How different can the bias of two polynomials be?
I'm trying to figure out how to approach the following question:
Let $g,h$ be polynomials over $\mathbb{Z}_p$ (for prime $p$) with $n>1$ variables.
Denote by $bias(g)=|\sum_{x\in \mathbb{Z}_p^n}e^{...
5
votes
0
answers
97
views
Rational functions with trivial Weil symbols at every point
Let $f, g$ be a pair of nonzero rational functions in $\mathbb{C}(t).$ For $\lambda\in \mathbb{C}$ let $a$ be multiplicity of $g(t)$ at $\lambda$ and $b$ - multiplicity of $f(t)$ at $\lambda.$ Weil ...
5
votes
0
answers
121
views
Is integer circuit membership undecidable?
According to wikipedia
integer circuit
in its simplest form is succinct representation of multivariate polynomial with
integer coefficients. Decidability if an integer is represented by the integer ...
5
votes
0
answers
113
views
Progress on the result about montonicity of Kazhdan Lustzig polynomials
I am reading the paper Masato Kobayashi---Combinatorics on Bruhat Graphs and
Kazhdan-Lusztig Polynomials.
Let $P_{x,w}$ be the Kazhdan Lusztig polynomial of $W$.
There is a result about ...
5
votes
0
answers
459
views
The Riemann zeta function and differential operators
I've revisited an old post of mine--Dirac's Delta Functions and Riemann's Jump Function J(x) for the Primes--dealing with Riemann's "jump" or "staircase" function (aka, Π(x)) that ...
5
votes
0
answers
162
views
Example of applying real quantifier elimination algorithm for polynomials
Sorry if any of this is unclear, or doesn't make much sense, I'm still trying to figure it out, a practical example such as this would likely help me understand better than anything else. I have read ...
5
votes
0
answers
193
views
A non-commutative analog of a result concerning a Jacobian pair
Let $k$ be a field of characteristic zero and let $E=E(x,y) \in k[x,y]$.
Define $t_x(E)$ to be the maximum among $0$ and the $x$-degree of $E(x,0)$.
Similarly, define $t_y(E)$ to be the maximum among $...
5
votes
0
answers
146
views
Descending chain of subalgebras of $k[x,y]$
Let $k$ be a field of characteristic zero.
Let $\{R_i\}_{i \in \mathbb{N}}$ be a descending chain of $k$-subalgebras of $k[x,y]$: $k[x,y]=:R_0 \supseteq R_1 \supseteq R_2 \supseteq \ldots$,
such that ...
5
votes
0
answers
124
views
How good are these probabilistic algorithms for the NP-hard problem gcd of sparse polynomials?
The paper NEW NP-HARD AND NP-COMPLETE POLYNOMIAL AND INTEGER DIVISIBILITY PROBLEMS David A. PLAISTED”
defines sparse polynomial as set $\{(a_i,i)\}$ and $f=\sum a_i x^i$.
On p.5:
Theorem 3.3. The ...
5
votes
0
answers
200
views
Resultant of a binomial and a trinomial
Does anyone know of any papers dealing with the resultant of a binomial $x^n+a$ and a trinomial $x^r+bx^s+c$ ?
Even special cases would be of interest.
(The resultant of two binomials is well known.)...
5
votes
0
answers
139
views
On a family of polynomials
Investigating experimentally a topic (somewhat related to Bernoulli convolutions), I came across families of polynomials and I wonder whether they belong to some well-known family. A closely related ...
5
votes
0
answers
315
views
Is there a matrix with this specific quadratic determinant?
We have $\det M=(a+b)(c+d)$ where
$M=\begin{bmatrix}
a& 0& -1& 0\\
0& c& 0& -1\\
b& 0& 1& 0\\
0& d& 0& 1
\end{bmatrix}$ and $\det M'=(a'+b')(c'+d')$...
5
votes
0
answers
272
views
Can this set of equations be solved explicitly for algebraic curves?
In my recent work I stumbled upon a set of two equations. I'm interested in solving by eliminating auxiliary variable "$z$" and getting algebraic curve in terms of $x$ and $y$ given by the zero locus ...