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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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
2 answers
432 views

On the derivative of the Bernstein polynomial

$\newcommand\Z{\Bbb Z}\newcommand\De{\Delta}$For a natural $n$ and a function $g\colon\Z\to\Bbb R$, let $B_n g$ be the corresponding Bernstein polynomial, so that $$(B_n g)(x)=\sum_{k\in\Z} g(k)\binom ...
Iosif Pinelis's user avatar
2 votes
1 answer
338 views

Why is this polynomial factorizable? [closed]

I met a curious problem on factorizing a homogenerous polynomial of degree 9. Problem: Show that the following polynomial can be divided by $(a_1+a_2+a_3)$: \begin{align} &\quad\left| \begin{array}...
LichenSDU's user avatar
  • 187
4 votes
0 answers
67 views

Combinatorial interpretation of a pfaffian identity?

Let $n$ be a positive even integer. We introduce three types of skew-symmetric matrices, $A_{n+2}$, $B_n$ and $C_n$ (the subscript denotes the dimension of the matrices) in terms of the variables $z_1,...
eddy ardonne's user avatar
3 votes
0 answers
176 views

Do all polynomials (other than generalized cyclotomic polynomials) have the spaced polynomial property?

Anna Erschler just asked me a question that is posed as Question 1.2 in her recent preprint with J. Frisch and M. Rychnovsky. I am asking it here with her permission - since I find it interesting (...
H A Helfgott's user avatar
  • 19.3k
4 votes
0 answers
214 views

When $\langle u,v,w \rangle$ is a maximal ideal in $\mathbb{C}[x,y]$?

Let $u,v,w \in \mathbb{C}[x,y]$ and let $\langle u,v,w \rangle$ be the ideal generated by $u,v,w$. It is known that for two elements the following result holds: $\langle u,v \rangle$ is a maximal ...
user237522's user avatar
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5 votes
1 answer
151 views

Rational functions of order $3$

Let $p$ be a prime, let $\mathbb K$ a field with $\operatorname{char}(\mathbb K)=p$ and $\mathbb{K}(x)$ be the field of fractions of the polynomial ring $\mathbb{K}[x]$, i.e. $$\mathbb{K}(x)=\left\{\...
Mersn's user avatar
  • 51
7 votes
2 answers
513 views

Polynomials such that $|p(z)|\leq p(|z|)$

Let $p(x)=1+p_1x+p_2x^2+\cdots+p_nx^n$ be a polynomial with real coefficients and no positive zeros. Define $$\mu(x)=\frac{xp'(x)}{p(x)}, \hspace{3ex} \sigma(x)=x \mu'(x).$$ Many years ago, as part of ...
Valerio's user avatar
  • 397
11 votes
2 answers
709 views

What relationship is there between repeated roots of discriminants and orders of roots of the original polynomials?

Disclaimer: I asked this problem several days ago on MSE, I'm cross-posting it here. The title sounds like a high school problem, but (as a grad student not in algebra) it feels subtle/deep. ...
Harambe's user avatar
  • 225
-2 votes
0 answers
78 views

System of polynomial equations and its Jacobian determinant

double post https://math.stackexchange.com/questions/4875170/system-of-polynomial-equations-and-its-jacobian-determinant Does this propostion hold? Proposition Let $\mathbb{C}[x_1,...x_n]$ be a ...
George's user avatar
  • 449
2 votes
1 answer
93 views

Bound for the $n$-th derivative of a proper rational function with no poles on the right half-plane

Suppose that $f$ and $g$ are polynomials with nonnegative coefficients, the degree of $g$ is greater than the degree of $f$, $g + f$ have no zeros on the right half plane $\mathbb{C}_+ = \{z \in \...
xen's user avatar
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Does a constant $C>0$ exist such that for $\forall\ p\in\mathbb{C}[z_1,z_2]$ we have: $\sup_{z\in rD^2}|p(z_1,z_2)|\le C\sup_{z\in D^2}|p(z_1,z_2)|$?

Question: Does a finite constant $C>0$ exist such that for $\forall\ p\in\mathbb{C}[z_1,z_2]$ we have: $$\sup_{z\in r \mathbb D^2}|p(z_1,z_2)|\le C\sup_{z\in \mathbb D^2}|p(z_1,z_2)|$$ where $r>...
anon's user avatar
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3 votes
1 answer
135 views

Solving a recursion for polynomials defined by a matrix product

Define the polynomial $p_n(X) \in \mathbb{Z}[X_1,...,X_n]$ as the top left entry in $A^n$ for the $(d \times d)$ matrix \begin{align*} & A = \left(\begin{matrix} X_1 & \dots & \...
Tardis's user avatar
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1 vote
0 answers
126 views

What is the possible reminders modulo 4 of an "odd part" of a polynomial?

Let $f(x)$ be a polynomial with integer coefficients. Let $f(x) = 2^k \cdot m$ where $m$ is odd. The questions are What are the possible values of $m \mod 4$ (1, 3 or both)? I want the algorithm ...
Denis Shatrov's user avatar
6 votes
1 answer
182 views

What is the smallest $\mathbb{Z}[x]$ multiple of $(x-1)^n$ in the coefficient vector $\ell_1$ sense?

Define a norm $\lVert p \rVert_1$ for $p\in \mathbb{Z}[x]$ as the sum of absolute values of the coefficients of $p$, as expressed in the ordinary monomial basis. What is the smallest norm of a ...
vujazzman's user avatar
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4 votes
0 answers
96 views

Finding the paper "Polynomial Inequalities" by Borislav Bojanov

I'm looking for the paper B. D. Bojanov, Polynomial inequalities, in “Open Problems in Approximation Theory” (B. Bojanov, Ed.), pp. 25–42, SCT, Singapore, 1993. The above reference is taken from the ...
Simon's user avatar
  • 81
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0 answers
87 views

On polynomial equation of fourth order depending on two parameters and bound on a maximal root

I would like to apologize in advance for a too technical question. Let us consider the following fourth order polynomial equation in $x$: \begin{eqnarray} F(x) \equiv (2 a p +2)x^4+ (6a(1-a)p^...
Vladimir's user avatar
  • 359
2 votes
1 answer
88 views

Does a polynomial $P(X,Y)$ that specializes to a polynomial $P(x_0,Y)$ with distinct roots in $\overline{k}$ have distincts roots in $\overline{k(X)}$

Let $k$ be a field, $P\in k[X][Y]$ be a monic polynomial of degree $n$ in $Y$. I am looking for a simple proof of the following fact. "If there exists $x_0\in k$ such that $P(x_0,Y) \in k[Y]$ has ...
Oblomov's user avatar
  • 2,501
0 votes
0 answers
63 views

Hensel lifting of roots of a biquadratic polynomial

Let $5$ divide $p-1$. Therefore, we have $$1+x+x^2+x^3+x^4=(x-\alpha)(x-{\alpha}^2)(x-\alpha^3)(x-\alpha^4)=f_1f_2f_3f_4$$ over $F_p,$ where $\alpha$ is an element of order $5$ in ${F_p}^\times.$ We ...
HIMANSHU's user avatar
  • 381
5 votes
1 answer
136 views

Upper bound for the $n$-th derivative of a rational function $\frac{f}{f+g}$

Let $f$ and $g$ be real polynomials with nonnegative coefficients. Let $$ h = \frac{f}{f+g}. $$ I want to prove that the $n$-th derivative of $h$ satisfies: There exists $C > 0$ such that $$ |h^{(...
xen's user avatar
  • 187
0 votes
1 answer
143 views

Coefficients of 0,1-polynomials factorization

Let $n\in\mathbb{N}$ and $P_{n}$ is $0,1$-polynomial whose coefficients are binary digits of $n$. Let $Q_{1}(x) \cdot Q_{2}(x) \cdots Q_{m}(x)$ - polynomial factorization (over integers) of $P_{n}$. ...
Denis Ivanov's user avatar
3 votes
0 answers
238 views

On thickness of binary polynomials

OEIS A169945 introduces the concept of thickness of a polynomial as the magnitude of the largest coefficient in the expansion of the square of the polynomial. Considering the $2^{n+1}$ polynomials $p(...
Sayan Dutta's user avatar
25 votes
1 answer
954 views

Is every polynomial of the form $2x^{2n} -x^n +1$ irreducible over $\mathbb{Z}$?

Is every polynomial of the form $2x^{2n} - x^n +1$ irreducible for $n>0$? Motivation: A few years ago a student asked if $29$ was the largest number which is prime and one more than a perfect ...
JoshuaZ's user avatar
  • 6,090
0 votes
0 answers
66 views

Quick calculation of a symmetric product with two indices

Say I have a product $\prod_{1\le i \le N-1}\prod_{i<j\le N-1} (1+t_i t_j a_{ij})$, where $a_{ij}$s are real number. I want to calculate the coefficient of $\prod_{0 \le i < N} t_i$. Is there an ...
pallab1234's user avatar
1 vote
1 answer
128 views

common zeroes of multivariable polynomials

Let $P_1(X,Y),\cdots,P_n(X,Y)$ be polynomials of $\mathbb C[X,Y]$ not all zero and $S$ be an infinite subset of $\mathbb C^2$ such that $P_1,\cdots,P_n$ vanish on $S$. My question: do there exist a ...
joaopa's user avatar
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0 votes
1 answer
38 views

relations between non-negativity of multivariate polynomials and SOS over gradient ideal

We know there is a necessary condition for the non-negativity of multivariate polynomials in the paper "Sum of Squares Decompositions of Polynomials over their Gradient Ideals with Rational ...
Werther's user avatar
  • 59
2 votes
1 answer
133 views

Existence of an integer coefficients polynomial with prescribed bounds on [0,4]

Is there a polynomial f with integer coefficients that satisfies the following criteria: f is not constant; for all $x\in[0,1]$, $1-\frac{1}{x}\leq f(x)\leq \frac{1}{x}$; For all $x\in [1,4]$, $0\leq ...
Yanlong Hao's user avatar
1 vote
1 answer
92 views

Extension of eigenvalue and eigenprojection to its complex neighbourhood for constantly hyperbolic operator

I am reading a paper named The block structure condition for symmetric hyperbolic systems written by G. Metivier. There is a statement really has bothered me for a long time. Consider the following ...
vent de la paix's user avatar
3 votes
1 answer
298 views

Rationals polynomials with integers values

I have asked this question here (*), but there are no answer. Let $q \in \mathbb N \cap [2,+\infty[$ and $P \in \mathbb Q[x]$ with $\forall k \in [0,\deg(P)] \cap \mathbb N, P(q^k) \in \mathbb Z$. Is ...
Dattier's user avatar
  • 3,737
0 votes
1 answer
109 views

Loss of degree for polynomials

Let $q$ be a power of a prime $p$, $m$ be an integer greater than $0$. Does there exist polynomials $P_0,\cdots,P_m$ of $\mathbb F_q[T]$ not all in $\mathbb F_q$ such that there exist polynomials $Q_0,...
joaopa's user avatar
  • 3,657
4 votes
0 answers
120 views

Vanishing exponential sums of fractional parts of polynomials

Let $p$ be an integer polynomial and $k$ be a natural number, both fixed. Is it the case that if $$C(\alpha) = \sum_{i=1}^k e(\alpha \{p(i)/k\})$$ equals 0, then $\alpha$ is an integer? Here, $e(x) = ...
Borys Kuca's user avatar
0 votes
0 answers
103 views

How many isolated points can a degree $d$ planar curve have?

Let $p(x,y)\in\mathbb R[x,y]$ be a bivariate polynomial of degree $d$. What is the maximum possible number of its acnodes (i.e. isolated roots in $\mathbb R^2$ not counting multiplicities)? A pretty ...
Fei's user avatar
  • 1
2 votes
0 answers
59 views

Eulerian polynomial from Bruhat interval - h* of something?

Let $\sigma \in S_n$ be a fixed permutation. Consider the polynomial $$ P_{\sigma}(t) = \sum_{\substack{\pi \in S_n \\ \pi \leq \sigma}} t^{\textrm{des}(\pi)} $$ where $\leq$ denotes Bruhat order, and ...
Per Alexandersson's user avatar
0 votes
1 answer
93 views

Necessary and/or sufficient condition for invertibility of the gradient of a polynomial of $m$ variables, viewed as a self map of $\mathbb{R}^m?$

I was wondering whether the following is true, and if not, is something known in this direction? Let $P:\mathbb{R}^m \to \mathbb{R}$ be a degree $r$ polynomial (not necessarily homogeneous) that ...
Learning math's user avatar
0 votes
0 answers
61 views

Reference Request: Factorization method for polynomials whose maximum absolute value of coefficient is 1

So, today I came up with a method for factoring polynomials whose coefficients are either $-1$ or $1.$ Let me explain with examples. Example No. 1. Factorize $P(x)=x^8+x^7+1$ Solution. It is known ...
Vanya Borisyuk's user avatar
1 vote
0 answers
44 views

generating set of polynomial ring

I am considering the polynomials $P=P[x_1,x_2,\ldots,x_n]$ with coefficients in a ring $R$. Consider a subset $S=\{p_1,p_2,\ldots,p_k\}$ of $P$. There is a map $f\colon P[x_1,x_2,\ldots,x_k] \to P$ ...
David Hillman's user avatar
0 votes
1 answer
130 views

On the solutions of system of homogeneous polynomials of degree $d$ in $n$ variables

Consider the following two system of n homogeneous polynomials in n variables of degree $d$ with complex coefficients: System 1 ($S_1$): $f_1(x_1,\dots,x_n) = 0$, $f_2(x_1,\dots,x_n) = 0$, $\vdots$ $...
GA316's user avatar
  • 1,219
0 votes
0 answers
58 views

Linear recurrences in coefficients of powers of quotients of polynomial rings

It is known that linear recurrences with constant coefficients can be computed via powers in $\mathbb{Z}[x]/f(x)$. We believe that this generalizes to quotients of multivariate polynomial rings. Let $...
joro's user avatar
  • 24.2k
4 votes
0 answers
138 views

Does an instance of this generalisation of the determinant exist?

Let $n$ be composite, $d$ a divisor greater than $1$ and $m=n/d$. Does anybody know if there is a general mapping $T$ from $n×n$ matrices to $m×m$ matrices that preserves the determinant? Over a field ...
Maarten Havinga's user avatar
0 votes
0 answers
102 views

Non-isomorphic cubic fields with a given discriminant

For a cubic field $K$ with defining polynomial $P(x)=x^3 + \frac{39}{25}x^2 + \frac{22}{25}x +\frac{4}{25}$ Magma calculates the discriminant $D=-3340$. ...
Maksym Voznyy's user avatar
4 votes
1 answer
340 views

Sufficient condition for a probability distribution on $\mathbb Z_p$ to admit a square-root w.r.t convolution

Let $p \ge 2$ be a positive integer, and let $Q \in \mathcal P(\mathbb Z_p)$ be a probability distribution on $\mathbb Z_p$. Question. What are necessary and sufficient conditions on $Q$ to ensure ...
dohmatob's user avatar
  • 6,706
2 votes
0 answers
65 views

Set partitions with big blocks - real-rooted polynomials?

The polynomials $$ T_n(t) := \sum_{\pi \in \text{Set Partitions}(n)} t^{\text{blocks}(\pi)} = \sum_{k=1}^n S(n,k)t^k $$ with $S(n,k)$ being the Stirling numbers of the second kind, are well-known to ...
Per Alexandersson's user avatar
1 vote
1 answer
227 views

Zeroes of elementary polynomials without involving closed-form solutions

Consider the following two polynomials, where $n$ is an integer: $$ p_n(x) = x^3-\frac1nx-\frac2n, \\ q_n(x) = x^2-\frac2n. $$ For any $n$, let $x_p=x_p(n)$ and $x_q=x_q(n)$ be the unique positive ...
chrisv's user avatar
  • 21
0 votes
1 answer
190 views

Simple question about 0,1-polynomials

Being interested in these polynomials, would like to clarify one small observation. Let $n\in\mathbb{N}$ and $P_{n}$ is $0,1$-polynomial whose coefficients are binary digits of $n$. Let $n$ has prime ...
Denis Ivanov's user avatar
0 votes
0 answers
176 views

Proof that the zeroes of certain polynomials are increasing with respect to degree

Choose $k+1$ positive integers $d_j\in\{0,1,2,3,\ldots\}$ and let $d=(d_1,\ldots,d_k)$. Consider the following polynomial equation over the positive reals: $$ \sum_{j=1}^{k}\frac1{x^{d_j}} = x^{d_{k+1}...
chrisv's user avatar
  • 21
10 votes
2 answers
463 views

Polynomial inequalities of the form $\int_0^1 |f(x)|^2 x \,dx \geq C \int_0^1 |f(x)|^2 \,dx$

Let $P_n$ denote all (real or complex) polynomials $f(x)=\sum_{k=0}^n a_k x^k$. I'm interested in inequalities of the form $$ \int_0^1 |f(x)|^2 x \,dx \geq C \int_0^1 |f(x)|^2 \, dx, \quad \text{for ...
Mfquing's user avatar
  • 141
18 votes
2 answers
1k views

Polynomials with many zeros of absolute value 1

Let $S$ be a finite subset of the positive integers. Define $N_S(x) = 1-(1-x)\sum_{j\in S}x^j$. Assume that $N_S(x)$ is symmetric, i.e., $x^dN_S(1/x)=N_S(x)$, where $d=\deg N_S(x)$. It seems that $N_S(...
Richard Stanley's user avatar
1 vote
0 answers
53 views

Factorization of the symmetric function identity $E(t)=1/H(t)$ with the refined Euler characteristic polynomials of associahedra / Lagrange inversion

I've come across two matrix identities, flagged with daggers below, relating the two sets of elementary and complete homogeneous symmetric polynomials/functions via the two sets of refined Lah and ...
Tom Copeland's user avatar
  • 9,897
0 votes
1 answer
61 views

Can non-periodic discrete auto-correlation be inversed?

I'm trying to understand whether discrete auto-correlation can be reversed. That is, we are given $t_0, \dots, t_n \in \mathbb C$ and a set of equations $$ t_{k} = \sum\limits_{i=0}^{n-k} b_i b_{i+k}, ...
Oleksandr  Kulkov's user avatar
36 votes
6 answers
3k views

Number of real roots of 0,1 polynomial

$0,1$ polynomial has coefficients from $\{0,1\}$. I investigate the number of roots in such polynomials. We are talking about real roots, and multiples are counted only once. It was found numerically ...
Denis Ivanov's user avatar

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