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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.

5
votes
1answer
130 views

Finding a particular matrix factor

Consider the following Laurent polynomial matrix-valued function in the variable $x\in\mathbb{C}$ $$ A(x) = \begin{bmatrix} 0 & x \\ x^{-1} & 0\end{bmatrix}. $$ I'm interested in finding a ...
1
vote
1answer
157 views

do you recognize this polynomial with double factorials?

I've got a polynomial (which comes from solutions of the heat conduction PDE) which seems so simple I'm wondering if anyone recognizes it $$f_{m}=x^{m-1} +(m-1)x^{m-3}+(m-1)(m-3)x^{m-5} +(m-1)(m-3)(m-...
4
votes
1answer
171 views

$L^1$ norm of Littlewood polynomials on the unit circle

A Littlewood polynomial is a polynomial with coefficients from $\{ 1, -1\}$ and the set of Littlewood polynomials with degree $n$ is denoted by $\cal{L}_n$. I'm interested in a "good" lower bound on ...
6
votes
1answer
240 views

An Optimization Problem with Complex Variables, regarding Eigenvalues of Circulant Matrices

Let $S$ be a finite subset of the complex unit circle and $1 \in S$. For each $n \in \mathbb N $, define $f_n\colon S^{n-1}\to\mathbb R$ by $$f_n(x) := \sum_{w^{n}=1}|x_1w+ x_2w^2\cdots+x_{n-1}w^{n-1}...
1
vote
0answers
76 views

Questions about polynomial systems with parameter

We fix $n \geq 1$. Let $f$ be a continuous function $f : \mathbb R_+^* \to \mathbb R^n$. Suppose that we have $n$ polynomials in $n+1$ variables $$\begin{aligned} P_1(Y, X_1, \dots, X_n)\\ P_2(Y, ...
1
vote
0answers
90 views

What can be said about $\{\deg(f),\deg(g),\deg(h)\}$, such that $k[f,g,h]=k[t]$?

Let $f=f(t),g=g(t),h=h(t) \in k[t]$, $k$ is a field of characteristic zero. Denote $\deg(f)=a$, $\deg(g)=b$, $\deg(h)=c$. Assume that $a \geq 2, b \geq 2, c \geq 2$. Assume that $k[f,g] \neq k[t]$, $...
3
votes
2answers
131 views

Factoring certain Hessians of real homogeneous bivariate polynomials

For any homogeneous polynomial $f \in \mathbb R [x,y]$, define the homogeneous polynomial $$H(f) := \partial_yf^2\partial_x\partial_xf-2\partial_xf\partial_yf\;\partial_x\partial_yf+\partial_xf^2\...
0
votes
0answers
79 views

Bounded polynomial having coefficients that are bounded linearly in degree and number of variables

Let $P(\mathbf{x})$ be a bounded multivariate polynomial of degree at most $d$ (for my purposes it can be either coordinate degree or total degree) over $[-1,1]^n$, and assume $|P(\mathbf{x})| \leq 1$....
8
votes
1answer
416 views

Prove that these are polynomials

Define the functions $$F_n(q)=\frac1{(1-q)^{2n}}\sum_{k=0}^n(-q)^k\frac{2k+1}{n+k+1}\binom{2n}{n-k} \prod_{j=0,\,j\neq k}^n\frac{1+q^{2j+1}}{1+q}.$$ The numbers $\frac{2k+1}{n+k+1}\binom{2n}{n-k}$ ...
1
vote
1answer
115 views

Dimension of $S$-units over $\mathbb{C}[x]$

Let $S=\{s_1,\ldots,s_n\}$ be a finite set of complex numbers. Consider the set of polynomials $$U=\{\,(x+s_1)^{k_1}\cdots(x+s_n)^{k_n}:\, 0\leq k_1+\cdots+k_n\leq H\}.$$ I am curious as to what is ...
2
votes
0answers
31 views

Lacunary fully reducible polynomial over a finite field

The following problem is motivated by this MO question on rich directions determined by a set of a finite plane. Problem Does there exist a constant $C$ such that for all odd primes $p$ there is a ...
7
votes
1answer
256 views

Real-rootedness of some polynomials

Denote the unsigned Stirling numbers of the first kind by $s(n,j)$. Question. Is it true that the polynomials $$P_n(x)=\sum_{j\geq0}s(n,j)\binom{x}j$$ have only real roots? Note. Obviously, ...
3
votes
0answers
60 views

Structure of $k[X,Y,X^a/Y^b]$, name for such rings

Rings of the form $k[X,Y,\frac{X^a}{Y^b}]$, where $k$ is a finite field and $a$ and $b$ are relatively prime natural numbers, are showing up as residue rings in a problem I'm studying, and I'm ...
10
votes
1answer
364 views

Real polynomial bounded at inverse-integer points

Let $p$ be a real polynomial and $N$ be a positive integer. Suppose I tell you that $|p(\frac{1}{k})| \le 1$ for all $k\in\{1,\ldots,N\}$, and also that $p(\frac{1}{N})\le -\frac{1}{2}$ while $p(\...
7
votes
0answers
155 views

Positivity of certain polynomial coefficients

Consider the rational functions (in fact, polynomials) $$F_n(q)=\frac1{(1-q)^{2n}}\sum_{k=0}^n(-q)^k\frac{2k+1}{n+k+1}\binom{2n}{n-k} \prod_{j=0,\,j\neq k}^n\frac{1+q^{2j+1}}{1+q}.$$ The numbers $\...
8
votes
0answers
142 views

Nonzero subdeterminants conjecture: has anybody seen this anywhere?

I already posted this question on Mathematics StackExchange. A user there suggested that I rather post it on mathoverflow, since it is a research question. So here it is. Let $m\geq2$, $n\geq1$ be ...
2
votes
0answers
155 views

Real-rooted polynomials with coefficient constraints

My question is whether there exists $(a_0, a_1, \ldots, a_{2n-1}) \in \mathbb{R}_{+}^{2n}$ such that (1). $a_{2k} + a_{2k+1} = \binom{3n-1}{3k} + \binom{3n-1}{3k+1} + \binom{3n-1}{3k+2}$ for all $0 \...
2
votes
1answer
220 views

Lüroth theorem for $k \subset k(f,g) \subseteq k(x)$

Let $k$ be a field of characteristic zero (I do not mind to assume that $k=\mathbb{C}$, if things are easier in this case). Lüroth theorem says that a field $L$, $k \subset L \subset k(x)$ containing ...
8
votes
1answer
199 views

Patterns in roots of integer-coefficient polynomials

Below are shown two displays of all the roots of polynomials $$c_n x^n + c_{n-1} x^{n-1} + \cdots + c_1 x + c_0 \;=\; 0$$ with each coefficient $c_i$ an integer $|c_i| \le M$ (including $c_i=0$). No ...
2
votes
1answer
249 views

Linear difference inequality

It is well known how to find a solution for the following linear difference equation $$h_{m} = h_{m-1} + a \cdot h_{m-2}$$ Finding the roots $r_1$ and $r_2$ of $r^2 - r - a$, we have that the ...
5
votes
1answer
135 views

When is a linear combination of the elementary symmetric polynomials reducible?

Let $n\ge 2$ and consider the polynomial ring $\mathbb F [X_1,...,X_n]$, where $\mathbb F$ is a field. Let $e_j:=e_j(X_1,...,X_n)$ be the elementary symmetric polynomial of degree $j$ in $X_1,...,X_n$...
7
votes
2answers
142 views

How different can the constituents of an Ehrhart quasi-polynomial be?

Consider a $d$-dimensional convex rational polytope $P\subset\mathbb{Q}^d\subset\mathbb{R}^d$. Then, it's a standard fact that in general the function counting the number of lattice points inside the ...
5
votes
2answers
68 views

Univariate polynomial interpolation with restricted degrees

Let $D=\{d_1, d_2, \ldots, d_n\}$ be an integer set. I'd like to know if I can interpolate any collection of $n$ points $(x_1, y_1), (x_2, y_2), \ldots, (x_{n}, y_{n})$ by a polynomial whose degree ...
2
votes
1answer
297 views

Polynomials with no multiple root

Let $a,d$ be polynomials of $\mathbb Z[X]$ with $\deg a>\deg d\ge0$ and $P$ be a polynomial of $\mathbb Z[X]$. Consider an infinite sequence of integers $(\lambda_n)_n$. Can one assert there exists ...
3
votes
1answer
78 views

Locally nilpotent derivation on $A[X,Y]$ whose kernel is $A$; where $A$ is an affine $k$ domain, $char k=0$

Let $k$ be a field of characteristic zero and $A$ be a $k$-algebra. A derivation on $A$ is a $k$-linear map $D: A \to A$ such that $D(ab)=aD(b)+bD(a), \forall a,b \in A$. A derivation is called ...
0
votes
0answers
46 views

Change of polynomial eigenvalues between polynomials

Given the polynomial eigenvalue problem $$ p_t(z) = det ( P(z) + Q(t) ) = 0, $$ where $P(z) = \sum_{i=0}^k P_i z^i$ with $P_i \in \mathbb{C}^{n \times n}$ and $Q(t) \in \mathbb{C}^{n \times n}$. The ...
1
vote
0answers
112 views

Factorially closed, finitely generated $k$-sub-algebra of $k[X_1,X_2,X_3]$ , where $k$ is algebraically closed field of positive characteristic

Let $S$ be a sub-ring of a commutative ring with unity $R$. Then $S$ is called factorially closed in $R$ if $a,b \in R$ and $ab \in S\setminus \{0\} \implies a,b \in S$. Let $k$ be an algebraically ...
2
votes
2answers
418 views

These polynomials are always either even or odd [duplicate]

The falling factorials are $(x)_n=x(x-1)\cdots(x-n+1)$ with $(x)_0:=1$. Define the forward shift $E$ and the discrete derivative $\delta=E-1$, respectively, by $$Ef(x)=f(x+1) \qquad \text{and} \qquad \...
3
votes
1answer
143 views

Cancellation problem for Laurent polynomial rings and power series rings

Throughout, let $k$ be an algebraically closed field. For two $k$-algebras $A,B$ let us write $A \cong_k B$ to mean that $A,B$ are isomorphic as $k$-algebras. It is known that if $A$ is an integral ...
1
vote
0answers
46 views

Factorially closed, finitely generated $k$-sub-algebra $A$ of $k[X_1,…,X_n]$, where $n>3$, $k$ is algebraically closed of char $0$, $trdeg_k A=n-1$

Let $S$ be a sub-ring of a commuttaive ring with unity $R$. Then $S$ is called factorially closed in $R$ if $a,b \in R$ and $ab \in S \implies a,b \in S$. My question is : Let $k$ be an algebraically ...
4
votes
1answer
268 views

Two bivariate polynomials (or rational functions) that generate $\mathbb{C}(x,y)$

Let $f=f(x,y),g=g(x,y) \in \mathbb{C}[x,y]$, each of degree $\geq 1$, and $f,g$ are algebraically independent over $\mathbb{C}$ (= their Jacobian $\in \mathbb{C}[x,y]-\{0\}$). (1) Is there a ...
1
vote
2answers
107 views

What would be a standard reference for the formula of the discriminant of $f(t^d)$?

I've posted this to Math.SE about a month ago: Seems like $$ \Delta(a_0+a_1t^d+a_2t^{2d}+...+a_nt^{nd})=(-1)^{n\frac{d(d-1)}2}d^{nd}(a_0a_n)^{d-1}[\Delta(a_0+a_1t+a_2t^2+...+a_nt^n)]^d, $$ where $\...
7
votes
1answer
170 views

invertibility of matrix over free associative algebra

For a commutative ring $R$, a matrix $A \in M_n(R)$ is invertible iff $\det (A)$ is a unit in $R$. Is there a similar criterion to determine invertibility (having two-sided inverse) of a matrix over a ...
3
votes
1answer
84 views

Lagrangian interpolation at Chebyshev points - estimate on coefficients in monomic basis

First, let us fix some Notation: Let $n\in\mathbb{N}$ and $x_i=\cos(\tfrac{(i+1/2)\pi}{(n+1)})$, $i=0,\dots,n$, be the Chebyshev points. Let \begin{align}L_i(x)={\displaystyle\prod_{\substack{0\leq j\...
6
votes
1answer
314 views

The anti-symmetrization of a kind of polynomials in $\mathbb{Z}[x_1,x_2,\ldots,x_n]$

Let $n$ be a positive integer and $S_n$ be the symmetric group on $\{1,2,\ldots,n\}$. Let $\mathcal{A}_n$ be the anti-symmetrization operator on $\mathbb{Z}[x_1,x_2,\ldots,x_n]$ such that for any $f(...
0
votes
0answers
53 views

How to bound the total variation distance between two polynomials of Gaussian random variables?

Let $X$ be standard Gaussian random variable $X \sim \mathcal{N}(0, 1)$. Let $P_1$ and $P_2$ be two polynomials of degree $d$ with known coefficients. $$ P_1(x) = \sum_{i=0}^d a_i x^i, \\ P_2(x) = \...
32
votes
4answers
2k views

A family of polynomials whose zeros all lie on the unit circle

I had posted the following problem on stack exchange before. Suppose $\lambda$ is a real number in $\left( 0,1\right)$, and let $n$ be a positive integer. Prove that all the roots of the ...
5
votes
1answer
356 views

higher order analogues of sylvester's law of inertia?

Sylvester's law of inertia (here I quote wikipedia) If A is the symmetric matrix that defines the quadratic form, and S is any invertible matrix such that D = SAS^{T} is diagonal, then the number ...
8
votes
2answers
294 views

Coefficients of shifted Bernoulli polynomials

I stumbled across the following curious empirical properties of the Bernoulli polynomials $B_n(x)$. Can anyone provide a reference or proof? Let $k\in\mathbb{Z}$, $k\geq 2$. Then (empirically): The ...
0
votes
1answer
152 views

Prove a special form of Schur polynomial identity [closed]

Let $A_k$ be the $n\times n$ matrix defined by $$ A_k=\left[ \begin{array}{} 1 & x_1 & x_1^2 & \cdots & x_1^{n-2} & x_1^k \\ 1 & x_2 & x_2^2 & \cdots & x_2^{n-...
0
votes
1answer
127 views

Searching for matrices with some property

I don't know if this question is considered research-related. If not, I will move it to Math SE. I am searching for matrices with the property $$|A|_F^2 = \deg( \chi_A(t) ) = 2 \deg( m_A(t)), tr(A) ...
1
vote
1answer
292 views

A product of polynomials

Let $f(n)=1+x^n+x^{2n}+...+x^{n^2}.$ Let $p(x)$ be $1+x+x^2+x^5+x^7+...$ where the exponents are the pentagonal numbers. Let $a(n)$ be the sequence of integers such that the coefficients of the ...
2
votes
1answer
60 views

Efficient algorithm to compute resultants of sparse polynomials?

Consider two polynomials $f,g\in\mathbb{F}_2$ of degree $O(2^n)$, with the property that they are extremely sparse (say, only $O(n)$ of the coefficients are non-zero). Is there a way to calculate ...
8
votes
2answers
259 views

Let $f \in \mathbb{Z}[x]$. Does $\bar{f}$ have as many roots in $\mathbb{F}_p$ as $f$ has in $\mathbb{C}$ for infinitely many primes $p$?

Let $f \in \mathbb{Z}[x]$ be a nonconstant polynomial. Consider $\bar{f} \in \mathbb{F}_p[x].$ Let $\rho_p$ be the number of distinct roots of $\bar{f}$ in $\mathbb{F}_p$, and let $\rho$ be the number ...
5
votes
0answers
96 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 ...
0
votes
0answers
131 views

If $F(x_1,0,\ldots,0)=(x_1,0,\ldots,0)$, then $F$ is bijective?

Let $k$ be an algebraically closed field of characteristic zero and let $f_1,\ldots,f_n \in k[x_1,\ldots,x_n]$ have an invertible Jacobian, namely, the determinant ot their Jacobian matrix belongs to $...
6
votes
0answers
193 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 ...
1
vote
1answer
61 views

Polynomial Eigenvalue Problem with few non-zero coefficients

Let us define a diagonal matrix $\mathbf{D}(\lambda) = diag(\lambda^{m_1}, \dots, \lambda^{m_n})$ with $\lambda\in\mathbb{C}$ and positive integers $m_1, \dots, m_n$. The generalized characteristic ...
2
votes
1answer
166 views

Abhyankar-Moh embedding theorem without algebraic closedness

After asking this question, I figured out that I am also interested in the following related question: Is Abhyankar-Moh theorem 1.6 still valid if we remove the algebraic closedness assumption? ...
4
votes
0answers
57 views

The mapping degree of a map generated by quadratic polynomials

Consider a proper map $F: \mathbb{C}^n\to \mathbb{R}^{2n-1}(n\geq 2)$ such that each component of $F$ is given by a sesquilinear form on $\mathbb{C}^n$. This to say, each component of $F$ can be ...