# 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.

2,533
questions

-1
votes

0
answers

41
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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>...

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 & \...

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 ...

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 ...

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 ...

0
votes

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^...

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 ...

0
votes

0
answers

62
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 ...

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^{(...

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 $P_{n}$.
...

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(...

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 ...

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 ...

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 ...

0
votes

1
answer

30
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 ...

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 ...

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 ...

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 ...

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,...

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 ...

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) = ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 $f\colon P[x_1,x_2,\ldots,x_k] \to P$ ...

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 ($S_1$):
$f_1(x_1,\dots,x_n) = 0$,
$f_2(x_1,\dots,x_n) = 0$,
$\vdots$
$...

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 $...

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 ...

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 $D=-3340$.
...

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 ...

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 ...

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 ...

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 ...

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}...

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 ...

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(...

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 ...

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},
...

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 ...

-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

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 ...

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 ...

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 ...

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
...

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 ...

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 ...

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 ...

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 ...