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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How do you quickly determine which coefficients are greater than zero when multiplying two univariate positive polynomials?

Suppose that I have two polynomials with a degree of $n$, $A(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0$ and $B(x) = b_nx^n + b_{n-1}x^{n-1} + ... + b_0$ and the coefficients of these polynomials are ...
mtber75's user avatar
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4 votes
1 answer
410 views

Subsets $E$ of $\mathbb{F}_{p^k}$ with vanishing polynomial subset sums

The following question arose in some discussions recently as a misunderstanding of another problem. Question: Which subsets $E\subset \mathbb{F}_{p^k}$ satisfy the property that $ \sum\limits_{x\in E}...
Josiah Park's user avatar
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2 votes
1 answer
177 views

Bernstein bound on the number of roots of a rectangular multivariable polynomial systems

I would like to know what Bernstein's bound is on multivariate polynomial systems where there are more equations than unknowns. I have a "generic" zero-dimensional multivariable polynomial system with ...
alext87's user avatar
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1 vote
0 answers
246 views

Method of Coppersmith optimal for multivariate?

It is shown that Coppersmith method yields optimal integer root extraction for univariate polynomials in https://arxiv.org/abs/1605.08065 and a follow up work attempts this for bivariate polynomials ...
Turbo's user avatar
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3 votes
0 answers
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Independence of number fields generated by roots of Littlewood polynomials

Let $\mathcal{R}_d \subset \bar{\mathbb{Q}}$ be the set of all roots of degree $d$ polynomials with $\{-1,0,1\}$ coefficients and $$ c(d) := \min_{\substack{ \alpha, \beta \in \mathcal{R}_d \\ \alpha^...
Vesselin Dimitrov's user avatar
6 votes
1 answer
308 views

$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$ is always an integer

Does anyone know if the following problem has ever been studied? Let $a$ and $b$ be two real numbers and consider the polynomial: $$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$$ where $n$ is a ...
jack's user avatar
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1 vote
0 answers
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About Kazhdan Lusztig polynomial evaluating at q=1

Given $w\le w'$ (in Bruhat ordering), does $P_{x,w}(1)\le P_{x,w'}(1)$ (in usual ordering of $\mathbb{R}$), where $P_{x,w}(q)$ is the Kazhdan Lusztig polynomial?
James Cheung's user avatar
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1 vote
0 answers
190 views

Prime generating polynomials

Continuation to this previous question. According to Lemke-Oliver, an irreducible polynomial $G$ of degree $g$ with positive leading coefficient and $\Gamma_G\neq0$ (with $\Gamma_G$ a certain factor ...
Delmastro's user avatar
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9 votes
0 answers
244 views

Semi-primes represented by quadratic polynomials

According to Lemke-Oliver, irreducible quadratic polynomials $G$ with positive leading coefficient and $\rho(2)<2$, (where $\rho(m)$ denotes the number of incongruent solutions to the congruence $G(...
Delmastro's user avatar
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3 votes
0 answers
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Solutions to $w_x=CA_x$, $w_y=CA_y$ other than $w=f(A)$ and $C=f'(A)$?

Let $R \in \{\mathbb{C}[t^2,t^3], \mathbb{Z}[\sqrt{5}]\}$, and let $A=A(x,y) \in R[x,y]$ with $\deg(A) \geq 1$ (total degree). I wish to prove or find a counterexample to the following claim: If ...
user237522's user avatar
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3 votes
1 answer
260 views

Positive real root separation (v2)

(This is a follow-up question to Positive real root separation) Let $\beta\in(1,2)$ and $\gamma\in(1,2)$ be Galois conjugates of height 1. That is, there exists a polynomial $p$ with coefficients $-1,...
Nikita Sidorov's user avatar
9 votes
0 answers
326 views

Is this a possible strengthening of the Lehmer conjecture?

Here is another possible refinement of the Lehmer conjecture. For $\alpha \in \overline{\mathbb{Q}}^{\times}$, let $C_{\alpha} \subseteq \mathbb{Q}(\alpha)$ be the maximal cyclotomic field contained ...
Vesselin Dimitrov's user avatar
4 votes
1 answer
247 views

Positive real root separation

Let $\beta\in(1,2)$ and $\gamma\in(1,2)$ be Galois conjugates of height 1. That is, there exists a polynomial $p$ with coefficients $-1,0,1$ such that $p(\beta)=p(\gamma)=0$ (not necessarily minimal)....
Nikita Sidorov's user avatar
2 votes
0 answers
277 views

Asymptotics of Littlewood polynomials

Littlewood in [L] states several conjectures regarding asymptotics of polynomials with $\pm1$ coefficients. He considers the class $\mathscr F$ of polynomials of form $\sum^n\pm z^m$ and asks whether ...
BigM's user avatar
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3 votes
1 answer
315 views

Roots of anti-palindromic polynomial if coefficients are odd.

This is in continuation of the question asked in this earlier post here. Given an anti palindromic polynomial of degree $n$ with odd coefficients, does it have roots on the unit circle?
Student's user avatar
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6 votes
1 answer
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$f,g \in \mathbb{Z}[x,y]$ satisfying: $\operatorname{Jac}(f,g)=0$ and $f,g \notin \mathbb{Z}[h]$ for every $h \in \mathbb{Z}[x,y]$?

Is it possible to find $f,g \in \mathbb{Z}[x,y]$ (with $\deg(f),\deg(g) \geq 1$) such that the following two conditions are satisfied: (1) $\operatorname{Jac}(f,g)=f_xg_y-f_yg_x = 0$. (2) ...
user237522's user avatar
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1 vote
1 answer
208 views

Reference request for anti-palindromic polynomials.

I have come across a lot of papers that are written about the palindromic polynomials, however, I am recently interested in polynomials satisfying $$f(-x) = x^nf(1/x)$$ for $n\geq 1$ and for all $x\...
Student's user avatar
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2 votes
1 answer
177 views

For every table of interpolating nodes, there is a positive continuous function whose interpolating polynomials are not positive infinitely often

Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that ...
user521337's user avatar
  • 1,189
4 votes
1 answer
292 views

A Conjecture about the integral related to Chebyshev polynomial

I am interested in the following integral related to the Chebyshev polynomials $$I_{n,m}:= \int_0^\pi \left(\frac {\sin nx}{\sin x}\right)^{m} dx,$$ where $n,m\in \mathbb{Z}^+.$ It is easy to see ...
Jacob.Z.Lee's user avatar
0 votes
1 answer
167 views

For which $n$, can we find a sequence of $n+1$ distinct points s.t. the interpolating polynomial of every +ve continuous function is itself +ve

Fix an interval $[a,b]$. For which integers $n>1$, does there exist $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that for every continuous function $f:[a,b] \to (0,\infty)$, the ...
user521337's user avatar
  • 1,189
4 votes
1 answer
987 views

Orthogonal basis of polynomials?

Let us define the basis of polynomials given by: $$ \begin{array}\ P_0=1, \\ P_1=x, \\ P_2=x(x-1), \\ P_3=x(x-1)(x-2), \\ P_4=x(x-1)(x-2)(x-3), \ldots\\ \end{array} $$ I would like to know if this ...
fernando's user avatar
  • 303
22 votes
2 answers
643 views

Does every positive continuous function have a non-negative interpolating polynomial of every degree?

Let $f:[a,b] \to (0,\infty)$ be a continuous function. Then is it necessarily true that for every $n\ge 1$, we can find $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that the ...
user521337's user avatar
  • 1,189
2 votes
1 answer
129 views

$(x + y + z)....(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$ To find $P$

$$(x + y + z)(x + y\omega_n + z\omega_n^{n-1})(x + y\omega_n^2 + z\omega_n^{n-2})....(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$$ where $\omega_n$ is an nth root of unity. The ...
Awe Kumar Jha's user avatar
0 votes
0 answers
241 views

gcd of polynomial values

Suppose that $f$ and $g$ are two coprime polynomials in $\mathbb Z[x]$. I'm interested in any sort of upper bound on $gcd(f(a),g(a))$, in terms of the integer $a$. Are there any results of this type?...
user avatar
3 votes
1 answer
133 views

A geometric property about certain polynomials in two variables

Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$ where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last ...
Ali Taghavi's user avatar
4 votes
1 answer
239 views

Is there any Menelaus-type theorem for polynomials?

Consider $n+3$ polynomials of degree $n$, say $P_1(x) , ... P_{n+3}(x)$. In addition, consider that there are distinct (it is not necessary that all of them be distinct) numbers $x_{ij}$ for $1 \...
MR_BD's user avatar
  • 550
7 votes
1 answer
244 views

The determinant of a $4\times4$ matrix associated to some specific polynomial as follow

Let $f\in \mathbb{R}[x_1,x_2,x_3,x_4]$ defined by $$f_a(x_1,x_2,x_3,x_4)=\prod_{1\leqslant i<j\leqslant4}(x_i-x_j)^{2a_{ij}}$$ where $a=(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})\in \mathbb{N}^6$. ...
user173856's user avatar
  • 1,987
0 votes
1 answer
99 views

Algebraic independence of certain values implies algebraic independence of functions?

It is quite general and elementary question. Is it possible that some holomorphic functions $f_1,\cdots,f_m $ on a region $\Omega $ of $\mathbb C$ satisfies: Whenever $(f_1(z), \cdots, f_m (z)) $ is ...
LWW's user avatar
  • 653
8 votes
0 answers
214 views

Is there an approximate formula for the discriminant of a sparse polynomial?

Consider integer polynomials $P \in \mathbb{Z}[X] \setminus \{0\}$ of a degree $D \geq 1$ and without multiple complex roots. Let me introduce a notation $$ d(P) := \frac{1}{D} \log{|\mathrm{Disc}(P)|}...
Vesselin Dimitrov's user avatar
4 votes
1 answer
136 views

Polynomial expansions via prime-base digits

Fix a prime number $p$. If $n$ is a positive integer, then denote $$\text{$\omega_{p,k}(n):=\#$ of $k$'s in the $p$-ary expansion of $n$}$$ and the total sum of all its $p$-ary digits by $$\Omega_p(...
T. Amdeberhan's user avatar
3 votes
1 answer
237 views

Solving polynomial inequalities -- efficient Positivstellensatz on a computer

I have about twenty five (multilinear) polynomials $f_1(\mathbf{x}), f_2(\mathbf{x}), \dots, f_{25}(\mathbf{x})$ all in fifteen variables and I would like to decide if there is a $\mathbf{y} \in [0,1]^...
BPN's user avatar
  • 543
2 votes
0 answers
57 views

Classes of curves with "determinant-like operation"

Consider a motivating example: Let $E\in \mathbb{Q}[y][x]$ be of degree $n=2$ (in $x$) and separable when viewed as a member of $\mathbb{Q}[x,y]$. Therefore we can calculate it's roots in $\mathbb{Q}[...
Omer Rosler's user avatar
4 votes
0 answers
114 views

Name of a polynomial basis

Does anybody know if the set of polynomials $\{ 1,x,x^2+2,x^3+3x,x^4+4x^2+6,{x}^{5}+5\,{x}^{3}+10\,x,{x}^{6}+6\,{x}^{4}+15\,{x}^{2}+20,\ldots \}$ appear in the literature? I am curious what they are ...
Yusuf Gurtas's user avatar
4 votes
1 answer
126 views

On the linear factors of a polynomial obtained from the determinant of a matrix whose entries are related to Binomial expansion

Consider the polynomial ring $R=\mathbb C[x,y]$. Consider the matrix $A=\begin{pmatrix} x^5+y^5&5x^5&10x^5&10x^5&5x^5\\5y^5&x^5+y^5 &5x^5&10x^5&10x^5 \\10y^5&5y^5&...
user521337's user avatar
  • 1,189
2 votes
0 answers
95 views

Polynomials passing through points with tangential conditions

In corollary here http://math.mit.edu/~lguth/Exposition/erdossurvey.pdf on polynomial methods it is said "(Parameter counting) If $S\subset\mathbb F^n$ is a finite set, then there is a non-zero ...
Turbo's user avatar
  • 13.7k
9 votes
0 answers
340 views

Which polynomials in the minors of a matrix are invariant under conjugation?

$\newcommand{\Cof}{\operatorname{cof}}$ This is a cross-post. Let $1<k<n$ be a fixed integer. I am trying to understand "what can be said" about the $k$-degree minors of a linear map $T:V \to ...
Asaf Shachar's user avatar
  • 6,611
3 votes
0 answers
110 views

Relationship bewteen Kazhdan-Lusztig Vogan polynomial and Kazhdan-Lusztig polynomial

Let $W^I=\{w\in W: w^{-1}\Phi_I^+\subseteq \Phi^+\}$, $W_I$ be the Weyl group generated by $I$ and $w_I$ be the longest element in $W_I$ Let $M(\lambda)$ be the Verma module with highest weight $\...
James Cheung's user avatar
  • 1,855
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
2 votes
1 answer
102 views

About Kazhdan-Lusztig polynomial

Let $(W,S)$ be a Coxeter system. One can have the Kazhdan-Lusztig polynomial $P_{x,\ y}(q)$. Does $P_{x,\ y}(q)=P_{x^{-1},\ y^{-1}}(q)$ for all $x,y\in W$?
James Cheung's user avatar
  • 1,855
5 votes
1 answer
183 views

Parabolic Kazhdan-Lusztig polynomial coincide?

Let $(W,S)$ be a Coxeter system. For any subset $I\subseteq S$, we can have the parabolic Kazhdan-Lusztig polynomial $P_{x,w}^I(q)$ with respect to $I$. Now consider $I\subseteq J\subseteq S$. Both $(...
James Cheung's user avatar
  • 1,855
2 votes
1 answer
238 views

radical of a certain ideal of sixteen variable polynomial ring, generated by the entries of certain matrices

Consider the polynomial ring $R=\mathbb C[x_1,x_2,...,x_{16}]$, and set $$X=\begin{pmatrix} x_1 &x_2&x_3 &x_4\\ x_5&x_6& x_7&x_8\\x_9&x_{10}&x_{11}&x_{12}\\x_{13}&...
user521337's user avatar
  • 1,189
2 votes
1 answer
196 views

Guess (or upper bound) the general formula for a double sequence

Let $t,s \geq 0$ be integers. We have the following recursive formula: $$f(t+1,s) = f(t,s) + f(t,s-1) + \sum_{0\leq a,b,c \leq h(t):\\a+b+c = s-1}f(t,a)f(t,b)f(t,c),$$ where $$h(t) = \frac{1}{2}3^t -\...
Wuchen's user avatar
  • 505
1 vote
1 answer
431 views

Estimating the derivative of a polynomial on the unit circle

Let $P(z)=\sum_{k=0}^na_kz^k $ be a polynomial of degree $n$ and $z_k (1\leq k\leq n)$'s be $n$th roots of $-1$. Then when $\theta=0$ the inequality $$|P'(e^{i\theta})|\leq \frac{4}{n}\left|\sum_{k=...
Venkat's user avatar
  • 21
7 votes
2 answers
289 views

Bound on sum of coefficients of polynomials w.r.t a weighted integral

Fix $k\in\mathbb{N}$ and assume $f(x)$ is a real polynomial of degree $n$ such that we have the normalization $$\int_{-1}^1f(x)^2\,(1-x)^kdx=1.$$ I am interested in the optimal size of the sum of the ...
T. Amdeberhan's user avatar
6 votes
0 answers
214 views

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
19 votes
2 answers
748 views

How can I distinguish a genuine solution of polynomial equations from a numerical near miss?

Cross-posted from MSE, where this question was asked over a year ago with no answers. Suppose I have a large system of polynomial equations in a large number of real-valued variables. \begin{align} ...
David Zhang's user avatar
  • 1,292
1 vote
0 answers
343 views

Intersection condition for polynomial ring and maximal ideals

In ring theory, there is interest in a condition known as the intersection condition. There is a brief comment in McConnell-Robson along these lines: Consider the ring $R = k[x,y]$ where $k$ is a ...
user304582's user avatar
10 votes
1 answer
676 views

Roots and relation between polynomials and their derivatives

This is probably easy but it might be interesting. Here goes $\dots$ Let $P\in\mathbb{R}[x]$ be a polynomial of degree $n>2$ and $P'=\frac{dP}{dx}$. If $x_1, x_2, \dots, x_n$ are the roots of $P(x)...
T. Amdeberhan's user avatar
19 votes
1 answer
691 views

Counting real zeros of a polynomial

I recently came across a criteria to count the number of real zeros of a polynomial $P(x)$ with real coefficients. Unfortunately I cannot find the reference! The criteria is the following: Form the ...
Michael Griffin's user avatar
2 votes
0 answers
359 views

Locating "Polynomial interpolation in several variables" by Alexander and Hirschowitz

I'm looking for a digital copy of the already classical paper "Polynomial interpolation in several variables" by Alexander and Hirschowitz, which in particular solves the problem of the generic Waring ...
Jose Brox's user avatar
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