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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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,...
Peter O.'s user avatar
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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 ...
T. Amdeberhan's user avatar
6 votes
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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 $u\in\mathbb{C}$, $\alpha$ ...
Fll'Yissetat's user avatar
6 votes
0 answers
360 views

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 ...
Vincent Nesme's user avatar
6 votes
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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 ...
loup blanc's user avatar
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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 $\...
Kashyap Rajeevsarathy's user avatar
6 votes
0 answers
166 views

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 $C_L$ ...
Mare's user avatar
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6 votes
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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 ...
Mare's user avatar
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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 ...
paul Monsky's user avatar
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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 ...
Vesselin Dimitrov's user avatar
6 votes
0 answers
406 views

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 ...
Alin Tomescu's user avatar
6 votes
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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 ...
Julian Barathieu's user avatar
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262 views

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}...
T. Amdeberhan's user avatar
6 votes
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251 views

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 ...
user_lambda's user avatar
6 votes
0 answers
257 views

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-...
thomashennecke's user avatar
6 votes
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-...
Diane's user avatar
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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 ...
Johann Cigler's user avatar
6 votes
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 ...
Pablo's user avatar
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206 views

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 ...
Turbo's user avatar
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6 votes
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 $f_1, f_2, \ldots, f_m$ ...
jsliyuan's user avatar
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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+...
Greg Egan's user avatar
  • 2,852
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 $$\...
Rebecca J. Stones's user avatar
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 ...
eins6180's user avatar
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6 votes
0 answers
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. ...
Igor Rivin's user avatar
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6 votes
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3k views

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 $...
Troels Bak Andersen's user avatar
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_{\...
Guillaume Aubrun's user avatar
5 votes
0 answers
164 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} \...
Rachid Ait-Haddou's user avatar
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) =\...
user967210's user avatar
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 ...
Dimitri Koshelev's user avatar
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 ...
Student's user avatar
  • 5,008
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 ...
user avatar
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 ...
joro's user avatar
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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 $n$ ...
Mike Krebs's user avatar
5 votes
0 answers
366 views

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 ...
user334097's user avatar
5 votes
0 answers
211 views

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 ...
SashaP's user avatar
  • 7,027
5 votes
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 ...
Turbo's user avatar
  • 13.7k
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%...
Mare's user avatar
  • 25.8k
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^{...
GWB's user avatar
  • 181
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 ...
Daniil Rudenko's user avatar
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 ...
joro's user avatar
  • 24.2k
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 ...
James Cheung's user avatar
  • 1,855
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 ...
Tom Copeland's user avatar
  • 9,937
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 ...
Evan's user avatar
  • 51
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 $...
user237522's user avatar
  • 2,783
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 ...
user237522's user avatar
  • 2,783
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 ...
joro's user avatar
  • 24.2k
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.)...
Gary McGuire's user avatar
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 ...
Benoît Kloeckner's user avatar
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 $\det M'=(a'+b')(c'+d')$...
Turbo's user avatar
  • 13.7k
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 ...
Caims's user avatar
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