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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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
127 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
121 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
178 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
  • 163
4 votes
0 answers
93 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
84 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
85 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
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0 answers
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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
4 votes
1 answer
120 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
  • 155
0 votes
1 answer
137 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
845 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
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0 answers
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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
0 votes
1 answer
124 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
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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
128 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
91 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
296 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
108 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
  • 3,647
0 votes
0 answers
41 views

Interpretation of the rank of a system $h=(h_1,\dots,h_R)$ of forms when $R=1$ [closed]

I am reading a paper Liu and Zhao - On forms in prime variables. The paper establishes the minimum number of variables required by a general system of forms of degree $d\ge2$ to have solutions in ...
Anish Ray's user avatar
  • 311
4 votes
0 answers
116 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
24 views

Help to solve an optimization problem [migrated]

I have a simple optimizing problem, I already know the answer but I don't know How to prove it! Assuming we have a series of positive and real number like: $$ x_1, x_2, x_3, ..., x_n $$ We want to ...
Amir Hashemi'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
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2 votes
0 answers
57 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
92 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
125 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
137 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
339 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
  • 6,686
2 votes
0 answers
64 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
224 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
188 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
172 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
458 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
  • 9,877
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
-5 votes
1 answer
68 views

Application of Resultant in Computer Algebra [closed]

Can you guys give me some application of resultant in Computer Algebra, it will be amazing if you guys can give me some paper or book to read more. Thanks so much
Luật Trần Văn's user avatar
3 votes
1 answer
123 views

Homogeneous regular (= polynomial component) maps with odd degree and their being global homeomorphisms in dimensions higher than one?

Let $F:\mathbb{R}^m \to\mathbb{R}^m, F:=(F_1\dots F_m)$ be a regular map, i.e. with components $F_i$ that are polynomials. Assume further that each $F_i$ is an odd degree (say $d$) homogenous ...
Learning math's user avatar
0 votes
0 answers
70 views

On the multiplicative group of quotients of polynomial rings

Related to this. The $p+1$ factorization algorithm works over $\mathbb{Z}/n\mathbb{Z}[x]/f(x)$ and hopes $p+1$ to be smooth. We are trying to generalize this to multivariate case and also try to find ...
joro's user avatar
  • 24.2k
3 votes
1 answer
266 views

Where to find or how to prove that the ratio of two Bernoulli polynomials is increasing?

It is well known that the classical Bernoulli polynomials $B_j(t)$ are generated by \begin{equation*} \frac{s\operatorname{e}^{ts}}{\operatorname{e}^s-1}=\sum_{j=0}^{\infty}B_j(t)\frac{s^j}{j!}, \quad ...
qifeng618's user avatar
  • 828
3 votes
1 answer
72 views

Evaluating the generalized continued fraction obtained from the factorization of a bivariate polynomial equation

Happy New Year, MO community! We need someone expert in Generalized Continued Fractions (GCFs), with a deep knowledge of the GCFs’ convergence properties, to solve the following problem. PROBLEM ...
Monk's user avatar
  • 125
2 votes
0 answers
128 views

Parametrizing "ternary cubic equals a square"

I am interested in an equation of the form $$\displaystyle y^2 = f(x_1, x_2, x_3),$$ where $f \in \mathbb{Z}[x_1, x_2, x_3]$ is a ternary cubic form. In particular, I am looking for an analogue of the ...
Stanley Yao Xiao's user avatar
2 votes
0 answers
144 views

Large prime divisors of values of a polynomial, in a given residue class

Let $f(X) \in \mathbb{Z}[X]$ be an irreducible polynomial of degree $d \geq 2$. Let $q \in \mathbb{N}$ be an integer, and let $q \mathbb{Z} + r$ be a residue class that contains infinitely many primes ...
Jakub Konieczny's user avatar
0 votes
0 answers
119 views

Nilpotent elements of $(\mathbb{Z}/n \mathbb{Z})[x_1,...,x_m]/\langle f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)\rangle$

This is generalization of the univariate case and also related to open problem. Let $n,k,m,B>1$ be positive integers and $f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)$ be polynomials with ...
joro's user avatar
  • 24.2k
4 votes
0 answers
164 views

Nilpotent elements of $K=(\mathbb{Z}/n \mathbb{Z})[x]/f(x)$

This is related to an open problem. Let $n$ be integer and $f(x)$ polynomial with integer coefficients and set $K=(\mathbb{Z}/n \mathbb{Z})[x]/f(x)$. Let $S$ be the set of degree 2 nilpotent elements ...
joro's user avatar
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