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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.

4
votes
1answer
93 views

Closure of polynomials in $L^2_w$ with log-normal weight function

Consider the Hilbert space $L^2_w$ with scalar product $\langle f,g\rangle_w =\int_0^\infty f(x)g(x)w(x)dx$ where the weight $w$ is the density function of a log-normal distribution $$ w(x)=\frac{1}{\...
2
votes
1answer
223 views

A polynomial function on $\mathbb{R}^3$ whose all level sets are mutually non isometric Riemannian manifolds

Is there a polynomial function $P: \mathbb{R}^3 \to \mathbb{R}$ with the following property?: P does not have any critical value and for all $c \neq c'$, $f^{-1}(c)$ and $f^{-1}(c')$ are non ...
10
votes
2answers
726 views

Simple question about polynomials

Starting from a problem in combinatorics, I ended up with a very simple problem about polynomials, which, unfortunately, I am not able to solve. Say we work over $\mathbb C$. Fix $d>1$. Is it ...
0
votes
1answer
182 views

Relation between degree of root of determinant polynomial and rank of the matrix

Let $A=[a_{ij}]$ be an $n \times n$ matrix with $a_{ij}=f_{ij}(x_1,...,x_m)$ where $f_{ij}(x_1,...,x_m)$ is a polynomial in $m$ variables over a finite field $\mathbb{F}_q$. Let $rank(A)=n$. Now ...
0
votes
1answer
118 views

Are the inverses of a set of quadratic polynomials linearly independent?

Is a collection of reciprocals of monic reducible quadratic polynomials, that is functions of the form $$ \{ \left( (x-a_i)(x-b_i) \right)^{-1} \}_{i=1}^{k}, $$ linearly independent over a finite ...
3
votes
1answer
168 views

Linear homogenous polynomials that generates several quadratic polynomials

This is a generalization of this question. Let $P_1, \ldots, P_m$, $Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots,x_n]$ be linear homogenous polynomials. Let $f_1, \ldots, f_s$ be a homogenous ...
1
vote
1answer
186 views

About Frobenius Determinant Theorem

Finite group $G=\{x_1,x_2,...x_n\}$. Consider $G$'s multiplication table to be an $n\times n$ matrix $A$. Set $x_i=1$, $x_j=0$ ($j≠i$), $1≤i≤n$, then we get $n$ permutation matrices $S_i$ ($1≤i≤n$) s....
0
votes
1answer
111 views

Complexity of a polynomial

Denote complexity $C(n)$ to be minimum degree of the polynomial in $\Bbb Z[x]$ that maps even integers from $0$ to $2^n-1$ with even $2$-adic valuation to $>2^{n-1}$ integers and odd $2$-adic ...
8
votes
0answers
194 views

Unprovable integer identity involving exponentiation

I was thinking about Tarski's problem, and was wondering what happens if we have a theory $T$ with two sorts $N,Z$ with intended interpretations $\def\nn{\mathbb{N}}$$\def\zz{\mathbb{Z}}$$\nn,\zz$ ...
8
votes
3answers
300 views

Explicit formulas for invariants of binary quintic forms

I am looking for explicit formulas for the four basic invariants $I_4, I_8, I_{12}, I_{18}$ of a generic binary quintic form, either given in the shape $$\displaystyle F(x,y) = ax^5 + 5bx^4y + 10cx^...
2
votes
3answers
280 views

Linear homogenous polynomials that generates one quadratic polynomial

Let $P_1, \ldots, P_m$, $Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots,x_n]$ be linear homogenous polynomials. Let $f$ be a homogenous quadratic polynomial of degree $2$. Assume that for every $i$ and ...
4
votes
1answer
124 views

Degrees of polynomials vanishing to various orders on a set of points

Suppose $X$ is a finite set of points in $\mathbb C^n$. Let $d_r$ denote the minimum degree of a polynomial vanishing to order $r$ at each point of $X$. By linear algebra, we know find can find a ...
9
votes
0answers
222 views

Reference request: integral formula for $\sum_{\text{roots }\lambda}e^{-|\lambda|^2}$

Consider a polynomial $f(z)=c\prod_m(z-\lambda_m)\in\mathbb{C}[z]$. I am mostly interested in the case where this actually lies in $\mathbb{R}[z]$, but that is not essential. I wanted to find a nice ...
3
votes
1answer
204 views

What does this permutation polynomial look like?

What is the number of terms of the unique multilinear polynomial $f\in\Bbb F_2[x_{1,1},\dots,x_{n,n}]$ in $n^2$ variables such that $f$ vanishes only on matrices that are permutations? Are there good ...
1
vote
0answers
98 views

Algebraic properties of harmonic polynomials

While studying a unique continuation property of some determinants, I encountered the following problem. Let $p_{ij}$ be real or complex harmonic polynomials (with homogeneous real and imaginary ...
6
votes
1answer
96 views

Reference request: maximal ratio of different norms of polynomials

Let us consider polynomials as functions on $[0,1]$, and so define \begin{align*} \|f\|_2 &= \sqrt{\int_0^1f(x)^2\,dx} \\ \|f\|_\infty &= \max\{|f(x)|: 0 \leq x\leq 1\}. \end{align*} I am ...
3
votes
1answer
176 views

Is the following function a polynomial?

I am reposting the second question from here (after clarifying it) on the recommendation of user "GH from MO". Let $b_1,b_2,\dots$ be an enumeration of $\mathbb Q$. Question 2: Suppose I define $$...
1
vote
1answer
98 views

Multiplicity of roots of a fewnomial

It is easy to show that a complex polynomial with $N$ non-zero coefficients cannot have a non-zero root of multiplicity $N$ or more. Is there any standard name / reference for this fact? Also, ...
5
votes
1answer
332 views

(Variation of an old question) Are these functions polynomials?

This is a followup to the question here: How to show that the following function isn't a polynomial over Q?. As before, let $b_1,b_2,\dots$ be an enumeration of $\mathbb Q$. The question might ...
19
votes
3answers
2k views

How to show that the following function isn't a polynomial over Q?

Enumerate the rationals as $b_1,b_2,\dots$ and define the (set) function: $$f(x) = (x-b_1)^2 + (x-b_1)^2(x-b_2)^2 + \dots.$$ At any particular $x$, only finitely many terms are non zero so this is ...
-1
votes
1answer
217 views

Why can't this polynomial vanish except when $x+y=0,xy= 0$? [closed]

Show that for any $x, y \in \mathbb R$ with $x + y \neq 0,xy\neq 0$ $$p(x,y) := x^6-2 x^5 y+2 x^5-x^4 y^2-2 x^4 y+x^4+4 x^3 y^3+2 x^3 y-x^2 y^4-4 x^2 y^3-4 x^2 y^2+2 x^2 y-2 x y^5+6 x y^4+2 x y^3+y^6-...
3
votes
1answer
575 views

What does it mean polynomials share Newton polytope?

I have trouble understanding the connection between polynomials and Newton polytopes. I will try to make a short introduction to my problem and hope you will catch on. In the end I will ask questions. ...
6
votes
2answers
445 views

Is every square root of an integer a linear combination of cosines of $\pi$-rational angles?

For example, $\sqrt 2 = 2 \cos (\pi/4)$, $\sqrt 3 = 2 \cos(\pi/6)$, and $\sqrt 5 = 4 \cos(\pi/5) + 1$. Is it true that any integer's square root can be expressed as a (rational) linear combinations of ...
1
vote
2answers
146 views

How to prove the following polynomial does not have root of a special form?

I'm working on a special kind of graphs. To prove some uniqueness, I need to prove that the polynomial \begin{equation} x^{8}-7x^{6}+14x^{4}-8x^{2}+1 \end{equation} does not have any root of the form \...
2
votes
0answers
76 views

Can Davenport's estimate be extended to cubic polynomials with non-zero discriminant?

In 1961 Davenport showed that $H$ large enough there is a constant $c > 0$ such that $$ \sum \lvert D(P) \rvert^{-1/2} < c H^2 $$ where the sum is taken over the irreducible polynomials of ...
4
votes
2answers
184 views

Polynomials $P$ with integer roots near to $X^{\mathrm{deg}(P)}$

Let $d$ be a positive integer. My question is: can we then find a positive integer $r$ (dependent on $d$) with integers $\alpha_1, \alpha_2, ..., \alpha_r$ and $\beta_1, \beta_2, ..., \beta_r$ such ...
0
votes
0answers
62 views

Generically parameterize quartics in arbitrary dimension

Suppose $x=(x_1,\dots, x_d) \in \mathbb{R}^{d}$ for an arbitrary dimension $d$. Let $p(x_1,\dots, x_d)$ be a degree 4 polynomial and consider the quartic defined by $p(x_1, \dots, x_d)=0$. Is it ...
2
votes
2answers
117 views

Questions about interlacing polynomials

If we have a finite set of real-rooted polynomials of the same degree such that any two of them have a common interlacing then does it imply that this set has a common interlacer? Lemma $4.2$ (top of ...
0
votes
0answers
139 views

Does there exist a homogeneous polynomial $F$ whose partial derivatives satisfy the following inequalities?

Given $n \geq 2$, I would like to find either a homogeneous polynomial $F \in \mathbb{Q}[x_1, \ldots, x_n]$ of degree $d > 1$ with the following properties: $W = \{ \mathbf{x} \in \mathbb{R}^n : F(...
6
votes
1answer
208 views

If $f,g \in D[x,y]$ are algebraically dependent over $D$, then $f,g \in D[h]$ for some $h\in D[x,y]$?

This question asks: If $f,g \in k[x,y]$ are two algebraically dependent polynomials over an arbitrary field $k$, is it true that there exists a polynomial $h \in k[x,y]$ such that $f,g \in k[h]$, ...
9
votes
1answer
211 views

Explicit forms for the roots of Eulerian polynomials

Let $E_n(z)$ be the Eulerian polynomial $$E_n(z) = \sum_{\tau \in \mathfrak{S}_n} z^{\operatorname{des}(\tau)}$$ where $\mathfrak{S}_n$ denotes the set of all permutations of $\{1,\ldots,n\}$ and $\...
3
votes
0answers
123 views

Diagonalization of Hermitian Matrix Polynomials

I have a question on the decomposition of polynomial matrices. Suppose $A(\lambda) = \sum_{j=0}^L \lambda^j A_j$ is an $n \times n$ matrix of polynomials, which is Hermitian on the real axis $\lambda ...
9
votes
2answers
627 views

Semi group of polynomials which all roots lie on the unit circle

Let $X=\{f\in \mathbb{C}[z]\mid |z| \neq 1 \implies f(z) \neq 0\} $. The motivation for consideration of such an $X$ is the the concept of Lee-Yang polynomials. With the standard multiplication, $X$...
21
votes
3answers
3k views

Minimal polynomial of cos(π/n)

I know that $\cos(\pi/n)$ is a root of the Chebyshev polynomial $(T_n + 1)$, in fact it is the largest root of that polynomial, but often that polynomial factors. For example, if $n = 2 k$ then $\cos(\...
0
votes
1answer
52 views

Is the difference between polyfits for two data series equivalent to the polyfit of the difference between the two data series?

Suppose that we have two series of data points $a(x)$ and $b(x)$ with the same domain of definition for $x$, and we fit two polynomial functions $f(x)$ and $h(x)$ (of the same order $n$) to them, ...
15
votes
2answers
719 views

Is there an explicit expression for Chebyshev polynomials modulo $x^r-1$?

This is an immediate successor of Chebyshev polynomials of the first kind and primality testing and does not have any other motivation - although original motivation seems to be huge since a positive ...
5
votes
0answers
130 views

On a family of polynomials

Investigating experimentally a topic (somewhat related to Bernoulli convolutions), I came across families of polynomials and I wonder whether they belong to some well-known family. A closely related ...
4
votes
1answer
182 views

Proximity of solutions to system of degree two polynomials

Everything in this post is over the complex numbers. I would like to know if for every $\epsilon > 0$ there exists $\delta > 0$ such that the following holds for every $n$ and every $d$ which is ...
4
votes
2answers
349 views

Summation of double exponential series

Let $q \in (0,1)$ and consider the following summation: $$S(q,n) = \sum_{i=1}^n {q^2}^i$$ Is there a closed form expression or upper and lower bounds for $S(q,n)$? Specifically, I am looking for ...
2
votes
2answers
187 views

Minimal polynomial of a trigonometric number

I am trying to calculate the minimal polynomials of $h_{1}=-\cos(\pi/n)-\sqrt{\cos(2\pi/n)}$ and $h_{2}=-\cos(\pi/n)+\sqrt{\cos(2\pi/n)}$ when $n$ is odd. I think (and numerical calculations suggest ...
1
vote
2answers
267 views

Budan's theorem and real part of the roots

Budan's theorem gives an upper bound for the number of real roots of a real polynomial in a given interval $(a,b)$. This bound is not sharp (see the example in Wikipedia). My question is the ...
1
vote
0answers
39 views

Determining the Associated Sequence If Sheffer Conditions are not Met

A sheffer sequence $s_n(x)$ is formed by considering the generating function $$\sum_{k=0}^\infty s_k(x)\frac{t^k}{k!}=A(t)e^{xB(t)}$$ where $A$ is an invertible power series, and $B$ is a delta ...
5
votes
1answer
247 views

Polynomials on spaces of matrices

Let $\mathbb{P}^N$ be the projective space parametrizing $n\times n$ non-zero matrices modulo scalar multiplication, and let $\mathbb{P}^M\subset\mathbb{P}^N$ be the subspaces of symmetric matrices. ...
18
votes
5answers
2k views

What are the properties of this polynomial sequence?

Consider following polynomial sequence. $$\begin{cases}a_{-1}=0,~a_0=1, \\a_{n+1}=x \cdot a_n \pm a_{n-1}\end{cases}$$ Here $+$ or $-$ is taken in such a way that all coefficients in $a_n$ do not ...
5
votes
1answer
317 views

Stability of root-finding near the unit circle

It is stated in several sources on numerical analysis that the general problem of polynomial root-finding is ill-conditioned, but that it is well-conditioned if the roots are near the unit circle. (e....
11
votes
0answers
348 views

A congruence involving roots of unity

Let $f(x) \in \mathbb{Z}[x]$ and suppose $f(\omega^j) \in \mathbb{Z}$ for all $j= 1, \dots, n$ where $\omega = e^{2 \pi i/n}$ is a primitive $n^{\text{th}}$ root of unity. Computational evidence ...
3
votes
1answer
138 views

$g(a)$ divides monic $f(a)$ many times, is this possible?

Let $f,g \in \mathbb{Z}[x],\deg(f),\deg(g)>1$ and $f$ is monic. Assume $f$ and $g$ are coprime. For integer $a$ is it possible $g(a) \mid f(a)$ many times? Is it possible unbounded number of ...
11
votes
1answer
1k views

Do almost all systems of quadratic equations have solutions?

If I have a system of linear equations, $A x = c$, with $A$ an $n\times n$ complex matrix, it is relatively easy to see that the set of matrices $A$ for which there is no (complex) solution has ...
1
vote
2answers
59 views

Is there a feasible way to compute the number of steps between two sequences generated by a linear feedback-shift register?

Consider a full-period LSFR with a feedback polynomial of degree n. In the cyclic sequence generated by the LSFR, each n-bit sequence appears exactly once. Given two n-bit sequences, one can define ...
8
votes
0answers
151 views

The density of minimal polynomials

Let $K$ be a number field, i.e., a finite extension of $\mathbb{Q}$. By the primitive element theorem there exists $\alpha \in K$ such that $K = \mathbb{Q}(\alpha)$. Let $$\displaystyle S_K = \{\...